Multiple teams reportedly expressed recent interest in trading for Darnold.
The 23-year-old is coming off a poor campaign with New York, and the club changed its entire coaching staff for the 2021 season.
New head coach Robert Saleh, who was hired in January to replace Adam Gase, said "a lot of things" need to be done in order to evaluate the Jets' roster.
Darnold recorded a career-low 72.7 passer rating through 12 starts in 2020. The 2018 third overall pick has completed 59.8% of his pass attempts, averaging 213.1 yards per game with 45 career touchdowns and 39 interceptions. He's posted a 13-25 record as a starter.
Darnold has one year and roughly $9.8 million remaining on his rookie contract. A team trading for him now could also pick up the fifth-year option on his deal, which would cost about $25 million guaranteed for 2022.
The Jets own the second overall pick in this year's draft.