The Jets haven't budged from their initial offer that's reportedly in the range of $7 million to $8 million per season. Fitzpatrick, 33, reportedly is seeking twice that amount annually.
The sides have been engaged in the stalemate since February. Since then, the Jets have flirted with at least two free agents, Brian Hoyer and Robert Griffin III, with the latter eventually signing with the Cleveland Browns.
Jets general manager Mike Maccagnan reiterated last week that Fitzpatrick remains the team's priority, adding that "it may take some time" before it's resolved.
The Jets, who currently have quarterbacks Geno Smith and Bryce Petty on the roster, might also select a quarterback the NFL draft on Friday or Saturday. They took linebacker Darron Lee with their first-round selection Thursday.
Maccagnan, speaking near the conclusion of the first round, said he wasn't aware of the specifics of the Fitzpatrick report. He said the team's position hasn't changed; re-signing Fitzpatrick is their focus at quarterback.
"It's the same stance we had before," Maccagnan said. "We're focusing on the draft primarily these next few days, so we can help the team going forward."
After he was acquired in a trade from the Houston Texans in 2015, Fitzpatrick led the Jets to a 10-6 record, throwing 31 touchdowns (a team record) and 3,905 yards -- both career highs. He started every game, including the last nine with a torn tendon in his left (nonthrowing) thumb -- an injury that required in-season surgery.
Fitzpatrick's familiarity with offensive coordinator Chan Gailey's offense helped galvanize the offense, which finished 10th in total yards -- its highest ranking since 1998.
The feel-good story ended on a sour note, as Fitzpatrick threw three fourth-quarter interceptions in the season finale -- a crushing loss to the Buffalo Bills that cost the Jets a playoff spot. Fitzpatrick has yet to reach the postseason in 11 years.
ESPN's Rich Cimini contributed to this report.