Financial terms were not disclosed by the team, but the NFL Network's Ian Rapoport reported the deal was worth $6 million.
McCown, who turns 38 in July, met with the team over the weekend. He also visited with the Dallas Cowboys, but Kellen Moore was re-signed to serve as Dak Prescott's likely backup.
McCown's immediate role with the Jets is in question with last year's starter Ryan Fitzpatrick unsigned. Backup Geno Smith, however, signed with the New York Giants on Monday while third-year quarterback Bryce Petty struggled in four starts in 2016, throwing seven interceptions before a torn labrum ended his season.
Injuries are nothing new for McCown, who was limited by a shoulder ailment in 2016. He passed for 1,100 yards with six touchdowns and six interceptions in five games with the Cleveland Browns.
McCown has played for seven NFL teams and has passed for 14,242 yards, 79 touchdowns and 69 interceptions in his career.