Russian troops have entered the outskirts of the Ukrainian city of Severodonetsk, Luhansk region governor Serhiy Gaidai said Monday, describing "very fierce" fighting in the ruins of a city that has become the focus of Moscow's offensive.

Russia has concentrated its firepower on the last major population center still held by Ukrainian forces in the eastern Luhansk province, in a push to achieve one of President Vladimir Putin's stated objectives after three months of war.

Incessant shelling has left Ukrainian forces defending ruins in Severodonetsk, but their refusal to withdraw has slowed the massive Russian offensive across the Donbass region.

Gaidai said Russian troops had advanced into the city's southeastern and northeastern fringes. But he said Ukrainian forces had driven the Russians out of the village of Toshkivka to the south, potentially frustrating Moscow's push to encircle the area.

Moscow seeks to capture all of Ukraine's industrial Donbass region, and Severodonetsk is key to that. Fierce street fighting is underway in the city as Ukrainian defenders are trying to push the Russians out, Mayor Oleksandr Striuk told the Associated Press (AP) in a phone interview. Russian troops have advanced a few blocks toward the city center, he said.

"The number of victims is rising every hour, but we are unable to count the dead and the wounded amid the street fighting," the mayor added. He said 12,000 to 13,000 civilians left in the city that once held more than 100,000 are sheltering in basements and bunkers to escape the Russian bombardment.

Russian forces stormed Severodonetsk after trying unsuccessfully to encircle it, and Ukrainian President Volodymyr Zelenskyy has described the situation as "indescribably difficult." A Russian artillery barrage has destroyed critical infrastructure and damaged 90% of buildings. The mayor has estimated that 1,500 civilians in the city have died since the war began, from Russian attacks as well as from a lack of medicine or treatment.

At least five people died following strikes on civilian infrastructure in Ukraine's separatist-controlled city of Donetsk, Russian investigators said Monday, claiming the attack was carried out by Kyiv's forces.

"On May 30, Ukrainian security forces shelled the centre of the city of Donetsk. According to preliminary information, five civilians were killed, including a teenager born in 2009," Russia's Investigative Committee said as quoted by Russian news agencies.

It added that 16 people were injured in the attack that damaged three schools.

'Unconditional priority'

After failing to capture Kyiv in March, Russia announced that the focus of its "special military operation" was now to seize the entire Donbass region, consisting of two provinces, Luhansk and Donetsk, that Moscow claims on behalf of separatist proxies.

Russian Foreign Minister Sergei Lavrov on Sunday said the "liberation" of the Donbass was an "unconditional priority" for Moscow.

Capturing Severodonetsk and its twin city Lysychansk on the opposite bank of the Donets River would give Russia effective control of Luhansk province, a point at which the Kremlin might be able to declare some form of victory.

But by focusing its effort on a battle for the single small city – Severodonetsk housed only around 100,000 people before the war – Russia might be leaving other territory open to eventual Ukrainian counterstrikes.

The past few days have seen initial signs of a potential Ukrainian counteroffensive in the south, where Moscow is trying to consolidate its control of Kherson province, captured in the early weeks after it launched its invasion in February.

Kyiv says its forces pushed back Russian troops in recent days to defensive positions in three villages – Andriyivka, Lozove and Bilohorka – all located on the south bank of the Inhulets River that forms the border of Kherson.

The Institute for the Study of War think tank said this Ukrainian counterattack so far did not appear likely to retake substantial territory in the near term, but could disrupt Russian operations and force Moscow to reinforce the area.

Pressure on east

While Russia concentrated its efforts in the east, Ukrainian forces pushed back over the weekend in the southern region of Kherson, the country's military leadership said. Just to the north of the Kherson front, a suspected Russian strike damaged the center of the Ukrainian-held town of Novyi Buh overnight, the town council said on Telegram.

Russia said it had also struck a shipyard in Mykolaiv, a major Ukrainian-held port just west of Kherson.

Meanwhile, two people were injured following an explosion in the Moscow-controlled city of Melitopol in southeastern Ukraine, with local pro-Kremlin authorities said pinning the blame on Kyiv.

The Ukrainian government urged the West to provide more longer-range weapons to turn the tide in the war, now in its fourth month. Zelenskyy said he expected "good news" in the coming days.

A Ukrainian soldier on patrol in trenches near the town of Bakhmut, southwest of Severodonetsk, spoke of a nagging fear that his government could be drawn into negotiating an end to the conflict that would result in Ukraine losing territory.

"You know now what I'm most afraid of, now that the fighting is so intense, so tough?" Dmytro, a former English language teacher, told Reuters television. "That we would be told: That's it, stop it, we have a ceasefire."

"A negotiated settlement can only happen on Ukrainian terms, and at present if it happened it would be a horror."