Folk had workouts with the Jets and Giants, but opted for the Jets deal.
It now is expected that it will be Folk's kicking job to lose this season and the Jets will allow Feely to test the free-agent market and leave if he finds a sufficient offer.
However, Feely says he'd prefer to stay with the Jets and sign a contract before the free-agency period begins.
"I'd love to get back there," Feely told ESPN.com's Tim Graham. "We have had negotiations and discussions, and they were very pleasant, very positive. They're very pleased with how I kicked over the past couple years and the leadership I showed. ... It's just a matter of coming to an agreement on a contract."
Last season, Feely made 30 of 36 field goal attempts and scored 122 points. He has 231 field goals and 1,013 points in nine career seasons.
Folk made 18 of 28 field goals and scored 90 points in 14 games last year. The Cowboys cut him with two weeks remaining in the regular season.
The Jets also signed free-agent offensive tackle Michael Turkovich.
Turkovich, a former Notre Dame standout, was in training camp with the Cowboys as an undrafted free agent last summer but was released after injuring a knee.
Adam Schefter is ESPN's NFL Insider. Information from The Associated Press was used in this report.