West was waived by the Chiefs on Wednesday after four seasons in Kansas City.
He gives the Jets seven running backs, including Bilal Powell, Isaiah Crowell, Elijah McGuire, Thomas Rawls, Trenton Cannon and George Atkinson III. Powell and Crowell are expected to share the bulk of carries in the regular season.
Cannon is a rookie sixth-rounder with versatility. McGuire is expected to miss at least several more weeks with a broken foot.
West was signed by the Chiefs in 2014 as an undrafted free agent out Abilene Christian. He has rushed for 999 yards and seven touchdowns in his career, along with 75 receptions for 552 yards and five TDs.