Ukraine's military warned the public on Tuesday of more indiscriminate Russian shelling of critical infrastructure as U.S. President Joe Biden issued one of his strongest warnings yet that Russia is considering using chemical weapons.

Biden, without citing evidence, said Russia's false accusations that Ukraine had biological and chemical weapons illustrated that President Vladimir Putin's "back was against the wall" and he was considering using such weapons.

"Now he's talking about new false flags he's setting up including, asserting that we in America have biological as well as chemical weapons in Europe, simply not true," Biden said at a business event. "They are also suggesting that Ukraine has biological and chemical weapons in Ukraine. That's a clear sign he's considering using both of those," he added.

The Russian Embassy in Washington did not immediately respond to a request for comment.

Biden also told businesses to be alert for cyberattacks by Russia. "It's part of Russia's playbook," he said in a statement.

The United States and its allies have previously accused Russia of spreading a claim that Ukraine had a biological weapons program as a possible prelude to using such weapons but Biden's remarks on Monday were some of his strongest on the subject.

Russia says it does not attack civilians although the devastation wrought on Ukrainian towns such as Mariupol and Kharkiv are reminiscent of previous Russian assaults on cities in Chechnya and Syria. Putin calls the war – the biggest attack on a European state since World War II – a "special military operation" to disarm Ukraine and protect it from "Nazis." The West calls that a false pretext for an unprovoked war of aggression.

Russian troops have failed to capture any major Ukrainian city more than four weeks into their invasion, and increasingly are resorting to massive destruction of residential areas with airstrikes, long-range missiles and artillery. The southern port of Mariupol has become a focal point of Russia's assault and is largely in ruins with bodies on the streets but attacks were also reported to have intensified on the second city of Kharkiv on Monday.

Russian forces were expected to continue to attack critical infrastructure with "high-precision weapons and indiscriminate munitions," said the Armed Forces of Ukraine in a statement.

Civilians making the dangerous escape from Ukraine’s embattled southern port hub of Mariupol described fleeing through street-to-street gunbattles and past unburied corpses as a steady Russian bombardment tried to pound the city into submission.

While Russian forces carried on with the siege after the city’s defenders refused demands to surrender, the Kremlin’s ground offensives in other parts of the country were advancing slowly or not at all, knocked back by lethal hit-and-run attacks by the Ukrainians.

The Ukrainian army said early Tuesday that it had forced Russian troops out of a strategically important Kyiv suburb following a fierce battle. The regained territory allowed Ukrainian forces to retake control of a key highway to the west and block Russian troops from surrounding Kyiv from the northwest.

But Ukraine’s Defense Ministry said Russian forces battling toward Kyiv were able to partially take other northwest suburbs, Bucha, Hostomel and Irpin, some of which had been under attack almost since Russia’s military invaded late last month.

Putin’s forces are increasingly concentrating their air power and artillery on Ukraine’s cities and the civilians living there, killing uncounted numbers and sending millions fleeing.

A senior U.S. defense official, speaking on condition of anonymity to discuss the military’s assessment, said Russia had increased air sorties over the past two days, carrying out as many as 300 in the past 24 hours, and has fired more than 1,100 missiles into Ukraine since the invasion began.

In a video address Monday night, Ukrainian President Volodymyr Zelenskyy hailed those who have fought back against Russia.

"There is no need to organize resistance," Zelenskyy said. "Resistance for Ukrainians is part of their soul," he added.

In Mariupol, with communications crippled, movement restricted and many residents in hiding, the fate of those inside an art school flattened on Sunday and a theater that was blown apart four days earlier was unclear. More than 1,300 people were believed to be sheltering in the theater, and 400 were estimated to have been in the art school.

Perched on the Sea of Azov, Mariupol has been a key target that has been besieged for more than three weeks and has seen some of the worst suffering of the war.

But no clear, independent picture emerged of how close its capture might be. Ukraine’s Defense Ministry said Tuesday that their forces were still defending the city and had destroyed a Russian patrol boat and electronic warfare complex.

Russia now controls the land corridor from Crimea, the peninsula it annexed in 2014, and is blocking Ukraine’s access to the Sea of Azov, the ministry said.

"Nobody can tell from the outside if it really is on the verge of being taken," said Keir Giles, a Russia expert at the British think tank Chatham House.

Over the weekend, Moscow had offered safe passage out of Mariupol – one corridor leading east to Russia, another going west to other parts of Ukraine – in return for the city’s surrender before daybreak Monday. Ukraine flatly rejected the offer well before the deadline.

Mariupol officials said on March 15 that at least 2,300 people had died in the siege, with some buried in mass graves. There has been no official estimate since then, but the number is feared to be far higher after six more days of bombardment.

For those who remain, conditions have become brutal. The assault has cut off Mariupol’s electricity, water and food supplies and severed communication with the outside world, plunging residents into a fight for survival. Fresh commercial satellite images showed smoke rising from buildings newly hit by Russian artillery.

Mariupol had a prewar population of about 430,000. Around a quarter were believed to have left in the opening days of the war, and tens of thousands escaped over the past week by way of the humanitarian corridors. Other attempts have been thwarted by the fighting.

Those who have made it out of Mariupol told of a devastated city.

"There are no buildings there anymore," said 77-year-old Maria Fiodorova, who crossed the border to Poland on Monday after five days of travel.

Olga Nikitina, who fled Mariupol for the western Ukrainian city of Lviv, where she arrived Sunday, said gunfire blew out her windows, and her apartment dropped below freezing.

"Battles took place over every street. Every house became a target," she said.

A long line of vehicles stood on a road in Bezimenne as Mariupol residents sought shelter at a temporary camp set up by Russian-backed separatists in the Donetsk region. An estimated 5,000 people from Mariupol have taken refuge in the camp. Many arrived in cars with signs that said "children" in Russian.

A woman who gave her name as Yulia said she and her family sought shelter in Bezimenne after a bombing destroyed six houses behind her home.

"That’s why we got in the car, at our own risk, and left in 15 minutes because everything is destroyed there, dead bodies are lying around," she said. "They don’t let us pass through everywhere – there are shootings."

In all, more than 8,000 people escaped to safer areas Monday through humanitarian corridors, including about 3,000 from Mariupol, Deputy Prime Minister Iryna Vereshchuk said.

Russian shelling of a corridor wounded four children on a route leading out of Mariupol, Zelenskyy said.

Elsewhere, Britain’s Defense Ministry said Ukrainian resistance has kept the bulk of Moscow’s forces more than 25 kilometers (15 miles) from the center of Kyiv, but the capital "remains Russia’s primary military objective."

In the Russian-occupied southern city of Kherson on Monday, Russian forces shot into the air and fired stun grenades at protestors who were chanting "Go home!" Kherson early this month became the first major city to fall to Russia’s offensive.

Ukrainian authorities also said Russia shelled a chemical plant outside the eastern city of Sumy, sending toxic ammonia leaking from a 50-ton tank, and hit a military training base in the Rivne region of western Ukraine with cruise missiles.

Russia’s invasion has driven nearly 3.5 million people from Ukraine, according to the United Nations. The U.N. has confirmed over 900 civilian deaths but said the real toll is probably much higher. Estimates of Russian deaths vary, but even conservative figures are in the low thousands.

Talks between Russia and Ukraine have continued by video but failed to bridge the chasm between the two sides. The Kremlin has demanded that Ukraine disarm and declare itself neutral. Zelenskyy told Ukrainian television late Monday that he would be prepared to consider waiving any NATO bid by Ukraine in exchange for a cease-fire, the withdrawal of Russian troops and a guarantee of Ukraine’s security.

Zelenskyy also suggested Kyiv would be open to future discussions on the status of Crimea, which Russia seized in 2014, and parts of the eastern Donbass region held by Russian-backed separatists. However, he said that was a topic for another time, after a cease-fire and steps toward security guarantees.

Diplomatic pressure

Biden is due to travel to Europe this week for meetings with allied leaders to discuss tighter sanctions on Russia, on top of the unprecedented financial penalties already announced. Ahead of the trip, he discussed Russia's "brutal" tactics in a call with European leaders on Monday, the White House said.

Russia's siege and bombardment of Mariupol, which European Union foreign policy chief Josep Borrell called "a massive war crime," is increasing pressure for action. But EU foreign ministers on Monday disagreed on whether and how to include energy in sanctions, with Germany saying the bloc was too dependent on Russian oil to impose an embargo.

Biden singled out India for being "somewhat shaky" in acting against Russia, its biggest supplier of military hardware, but praised the other members of the Quad group, Australia and Japan.

India has urged an end to the violence in Ukraine but has abstained from voting against its old Cold War ally Russia.

Even though India has grown close to the U.S. in recent years, it still depends on Russia for a continuous supply of arms and ammunition amid a Himalayan border standoff with China and perennial tension with Pakistan.

Kyiv seeks face-to-face meeting

While Biden was underlining the threats Putin may pose for Ukraine in the future, Zelenskyy renewed an offer of direct peace talks with his Russian counterpart late Monday, declaring the status of disputed territories could be up for debate and a possible referendum.

Zelenskyy told local media that he was ready to meet Putin "in any format" to discuss ending the almost one-month-old war that has shattered several Ukrainian cities. Zelenskyy said even the status of Russian-occupied Crimea and Russian-backed statelets in Donbass was up for negotiation.

"At the first meeting with the president of Russia, I am ready to raise these issues," he said. "There will be no appeals or historical speeches. I would discuss all issues with him in great detail" Zelenskyy said.

Russia has declared Crimea part of Russia and recognized the independence of the self-proclaimed Donetsk People's Republic (DNR) and Luhansk People's Republic (LNR) in eastern Ukraine.

All three areas were part of Ukraine following the collapse of the Soviet Union and are at the center of a decadelong crisis that on Feb. 24 spilled into an invasion and full-scale war.

"If I have this opportunity and Russia has the desire, we would go through all the questions," he told Ukrainian journalists in an interview published by media outlet Suspilne. "Would we solve them all? No. But there is a chance, that we partially could – at least to stop the war," he added.

Although Zelenskyy signaled that he was willing to talk about the status of the three areas, he has repeatedly insisted all three were part of Ukraine and that his country would not surrender. Zelenskyy also warned that any peace agreement involving "historic" changes would be put to a national referendum.

Sonia Mycak, a Ukraine expert at the Australian National University, said the promise of a popular vote likely dooms any suggestion of Kyiv ceding territory.

"The vast majority, like 80%, of Ukrainians are saying that they do not want to relinquish" those territories, Mycak said, citing two recent public opinion polls. "I think it would be rejected by the population, I really do. Very high numbers of Ukrainians are saying 'we should not stop fighting,'" she added.

"Ukrainians see themselves as under existential threat. It's not just the loss of territory it's the fact that they would have to live as Russians, there would be heavy Russification, there would be autocratic control," she explained.

A month of talks between Ukrainian and Russian officials have so far failed to stop or even slow a war that has forced 3.5 million Ukrainians to leave the country. But with Russia's much-larger military seemingly unable to occupy the entire country or topple Zelenskyy's ever-more-popular government, the Ukrainian leader said the war was inevitably going to end at the negotiating table.

"It is impossible not to have a solution. By destroying us, he is definitely destroying himself," Zelenskyy said of Putin. "I do not want us to go down in history as heroes and as a nation that does not exist ... And if they destroy themselves, they won't even have any heroism left," he added.