The veteran running back will likely serve in a third-down capacity if he makes the team.
Vereen recorded 164 rushing yards and 253 receiving yards in 2017 with the Giants, doing little to help an offense that floundered in a 3-13 season.
A seven-year veteran, Vereen spent the first four years of his career with the New England Patriots. He has never carried the ball more than 96 times in a season due in part to injuries and concern over his ability to handle more than a third-down role.
Though he has played 16 games in three of the past four seasons, the 29-year-old has taken part in only 79 of a possible 112 career contests.
He told Michael Eisen of the Giants' official website (via Pro Football Talk):
"I hate it when I have to sit out. I have been through a lot of training camps and sometimes I wake up and want nothing to do with it. But I am a much happier person when I am practicing and involved with the team and getting better and working on my craft than I am without. I like being with the team and getting days off when coach gives us days off."
The Saints will likely look for Vereen to fill a pass-catching role out of the backfield and take a handful of carries per game.
Vereen has caught at least 44 passes in four of the last five seasons, so he should be a fine low-cost addition. Expecting him to offer anything more than a rotational option—as some did following his breakout eight-game stretch in 2013—is probably futile.
It isn't guaranteed Vereen will be on the roster when the regular season begins thanks to New Orleans' depth at running back.
In addition to starter Alvin Kamara, the team has Terrance West, Trey Edmunds, Boston Scott, Daniel Lasco and Jonathan Williams.
Also, Mark Ingram will take on a big workload when he returns from a four-game suspension.
Although the Saints are a good fit for Vereen thanks to their ability to get running backs involved in the passing game, the competition could make it difficult for him to stand out.