Stewart was released by the Carolina Panthers at the end of last month, ending his 10-year tenure with the team.
The move reunites Stewart with new Giants offensive coordinator Mike Shula, who previously served in the same capacity with Carolina.
The contract cannot be finalized until the new league year opens on Wednesday at 4 p.m. ET.
Stewart, who will turn 31 on March 21, amassed 7,318 career rushing yards with 51 rushing touchdowns on 1,699 carries with the Panthers.
The 13th overall pick in the 2008 draft, Stewart rushed for 680 yards and a career-worst 3.4 yards per carry last season while sharing the backfield with Stanford rookie Christian McCaffrey.
In 2015, Stewart earned a Pro Bowl selection after leading Carolina with 989 rushing yards with six touchdowns on a career-high 242 carries. He helped the team set franchise records with an NFL-leading 500 points scored and 59 touchdowns while ranking second in the league with 142.6 rushing yards per game.
New York had one of the league's worst ground games last season, averaging 96.8 yards per game and producing six rushing touchdowns.