The Lions used the 35th overall choice to select the Georgia running back in the second round.
Detroit has signed five of its nine draft choices, with only first-rounder Jeff Okudah, fifth-rounder Quintez Cephus, fifth-rounder Jason Huntley and seventh-rounder Jashon Cornell yet to agree to terms.
Swift finished his career at Georgia with back-to-back, 1,000-yard seasons. He rushed for 1,218 yards and seven touchdowns as a junior.
The Lions have not had a 1,000-yard rusher since Reggie Bush in 2013.
Kerryon Johnson, who has missed 14 games in his two seasons, led the team with 403 yards in eight games last season and has said he was exited about sharing the backfield with Swift. Ty Johnson and Bo Scarbrough also are in the running backs room.