Johnson, 28, was released by the Titans after six 1,000-yard seasons in Nashville.
The Jets envision Johnson as a big-play threat who could pair with Chris Ivory when the offense slows to coach Rex Ryan's preferred "ground and pound" approach of relying on power running to move the chains and wear down opponents.
The Denver Broncos and Dallas Cowboys are also reportedly interested in signing Johnson in a timeshare arrangement at running back.
Johnson signed a four-year, $53 million contract with the Titans in 2011 that included a base salary of $8 million for the 2014 season. New coach Ken Whisenhunt and general manager Ruston Webster made it clear Johnson was not part of the long-term plans. Attempts to trade Johnson for draft pick compensation didn't materialize.
Johnson might not be the same player he was in 2009, when he rushed for more than 2,000 yards, but he carried the ball 279 times in 2013 and could flourish as part of a rotation. With the Jets, Ivory and Bilal Powell figure into the workload.
Johnson is also a solid short-range receiver with an average of 45 catches in Tennessee. That reliable outlet makes him the ideal check-down read for second-year quarterback Geno Smith, competing with Michael Vick to start for the Jets in 2014.