At 0805 GMT, the lira stood at 5.6900 against the U.S. currency, firming from a close of 5.7165 on Monday and making it the strongest performing emerging market currency against the dollar.
Broadcaster CNN Turk reported that U.S. ambassador to Ankara David Satterfield had recently given a presentation about the proposed package at the presidential palace prior to President Tayyip Erdogan's current visit to the United States.
The reports referred to an existing target of quadrupling bilateral trade between the two countries to $100 billion a year, an ambitious goal which has been mentioned in talks between Turkish and U.S. officials.
"The issue which would cause the biggest problem in U.S. relations would be Turkey turning to Russia for its plane needs after being removed from the U.S. F-35 project," said a treasury desk trader at one bank.
"The strengthening of reports that could break this vicious circle have had a positive impact on the lira this morning."
CNN Turk said the proposed deal was a free-trade accord including proposals regarding Turkey's participation in the F-35 stealth jet programme and the sale of U.S. Patriot missiles, as well as steps to lower customs duties on steel and aluminium.
Ankara and Washington have been at odds over Turkey's purchase of Russian S-400 missile defences, which has raised the prospect of U.S. sanctions and which Washington says pose a threat to Lockheed Martin Corp's F-35s.
U.S. Republican Senator Lindsey Graham said on Sunday after meeting Erdogan that he would like to get Turkey back in the F-35 joint manufacturing programme, from which it was suspended in July.
In an interview with Reuters on Sept. 13, Erdogan said he will discuss buying Raytheon Co's Patriot missile defence system with President Donald Trump this month.
Erdogan and Trump spoke by phone on Sunday, and may meet at the UN General Assembly in New York this week.
The main share index rose 1.5%, with the banking index up 2.1%. The yield on the benchmark 10-year bond fell to 14.61% from 14.82% on Monday.