Maye, poised to become an unrestricted free agent for the first time, is the first Jets player since defensive end Muhammad Wilkerson in 2016 to get the designation.
The franchise-tag amount for a safety is an estimated $10.5 million, which will be Maye's 2021 guaranteed salary if the two sides can't reach a long-term agreement by July 15. Maye, a second-round pick in 2017, made $6.6 million over his first four seasons.
Jets general manager Joe Douglas said last week the organization's goal is to lock up Maye with a new contract, but they apparently determined they wouldn't reach an agreement by Tuesday's 4 p.m. ET deadline. That deadline reportedly could get pushed back while the salary cap is finalized.
Maye's agent, Erik Burkhardt, was critical of the Jets last week on Twitter, saying the Jets "refuse to take care of their best player, captain and team-voted MVP."
Douglas insisted the acrimony will have no impact on the negotiations, saying the two sides have engaged in "productive conversations."
"Look, this is part of the business, part of negotiating," Douglas said. "I don't see this as affecting our ability to get something done with Marcus that's beneficial for both him and the team."
Maye, a four-year starter, blossomed last season as the undisputed leader of the New York secondary. Playing without safety Jamal Adams for the first time -- the unhappy Adams was traded to the Seattle Seahawks before training camp -- Maye recorded two interceptions, two sacks and two forced fumbles.
Known for his quiet, steady presence, Maye stepped out of character after a last-second, late-season home loss to the Las Vegas Raiders, openly criticizing defensive coordinator Gregg Williams for calling an all-out blitz on the Raiders' winning touchdown -- a tactic that seemed overly aggressive. The next day, Williams was fired.
The knock on Maye is his lack of interceptions -- only six over his first four seasons, which ranks tied for 28th among NFL safeties.