Multiple Turkish media outlets reported yesterday that the U.S. had offered Turkey a new deal on F-35 fighter jets, boosting optimism that strained relations between the two NATO allies were making progress.

U.S. envoy to Ankara David Satterfield reportedly offered Friday to sell Turkey the Patriot missile defense system, Habertürk and NTV reported.

The offer also included lower tariffs on steel and aluminum as well as a trade deal package that will help the two countries boost trade volume to $100 billion from the current $20 billion – a target Turkey's President Recep Tayyip Erdoğan and his American counterpart Donald Trump both agreed to reach during their G20 meeting in Osaka, Japan in June.

Reports also said a written version of Satterfield's offer is expected to be sent to the Turkish government.

In July, the U.S. suspended Turkey's involvement in the F-35 fighter jets program, saying the latter's purchase of the Russian S-400 missile defense system could endanger the aircraft, a claim Turkey has consistently denied.

Turkey produces some parts of the F-35 jets and is also a partner in the jet program. It has warned any effort to remove it from the production chain would be very costly.

After protracted unsuccessful efforts to purchase Patriot missiles from the U.S., Turkey signed a deal with Russia in April 2017 to acquire the Russian air defense system.

In response to U.S. concerns, Turkey has emphasized that the S-400 would not be integrated into the NATO systems and therefore had no chance to pose any threat to the alliance or its armaments. Turkey even proposed setting up a commission to clarify any technical issues. However, the U.S. has yet to respond to the proposal.

Tensions deepened in July when Turkey received its first shipment of Russian equipment. The delivery of the second battery of the system, which started on Aug. 27, was completed on Sept. 15.

Ankara said it was the U.S.' initial refusal to sell Patriot missiles that led it to seek out other sellers, adding that Russia offered it a better deal, including technology transfers.

The S-400 system is one of the most advanced air defense systems in the world, capable of simultaneously tracking multiple targets.

In the meantime, after a meeting with the Turkish president in New York Sunday, Republican Senator Lindsey Graham expressed hope for a more strategic relationship between Ankara and Washington.

"I am hopeful we can get a more strategic relationship with Turkey [...] try to get them back in the F-35 program, maybe talk about free trade program," Graham told reporters.

In his remarks after meeting Erdoğan on the sidelines of G20 summit in Osaka, President Trump had said the dispute as a whole was "unfair" to Turkey as Washington repeatedly turned down Turkey's requests to purchase Patriot missile systems in the past.

"Because Turkey bought a Russian missile, we're not allowed to sell them billions of dollars' worth of aircraft. It's not a fair situation," Trump had said.

He blamed the Obama administration's reluctance to sell Patriot missiles to Turkey in 2013, saying Turkey was not treated fairly as a NATO member.

The Turkish president arrived in New York on Saturday to attend the 74th session of the U.N. General Assembly. He is also expected to meet Trump on the sidelines of the assembly.