In a corresponding move, the Saints waived running back Daniel Lasco with a failed physical designation.
The move to add Vereen comes as the Saints brace to be without fellow running back Mark Ingram, who will begin the season by serving a four-game suspension for violating the league policy for performance-enhancing substances.
Vereen recorded 164 rushing yards and 253 receiving yards in 16 games with the New York Giants last season. He had 582 yards rushing and 842 yards receiving to go with five overall touchdowns during his three-year stint with the Giants.
The 29-year-old Vereen spent his first four NFL seasons with the New England Patriots. He set a Super Bowl record for receptions by a running back with 11 in the Patriots' victory over the Seattle Seahawks in Super Bowl XLIX.
Vereen will join a crowded running backs corps that includes Offensive Rookie of the Year Alvin Kamara, Terrance West, Trey Edmunds, Jonathan Williams and rookie Boston Scott.
Lasco, 25, did not have a carry while playing in three games with the Saints last season.