He is expected to be the team's starting fullback, a source told ESPN's Adam Schefter.
The Cardinals released running back Bronson Hill in a corresponding move.
Coleman, the NFL's first legally deaf offensive player, played for the Atlanta Falcons last season, appearing in all 16 games and starting seven.
He wears hearing aids in both ears and reads lips. He tries to position himself near the quarterback in the huddle so he can see his lips and get the play call. Asked last year by ESPN's Vaughn McClure how he hears the snap count, Coleman said: "I move when the ball moves.
"So as soon as we break the huddle, and let's say we have a protection going on, I'm looking, I'm looking, I'm looking, so by the time I'm in my stance, I already know where everybody is. I already know what I've got to do. And I'm looking at the ball. When the ball is snapped, I'm able to go. And I adjust.
"I've jumped a couple of times, all because I've been excited or because my mind has been going crazy. It's never because of my hearing loss. When that ball moves, that's when I move, plain and simple. If the ball doesn't move, I don't move."
Coleman, 27, spent the first four seasons of his career with the Seattle Seahawks and earned a Super Bowl ring.
He was charged with felony hit-and-run in October 2015. The charges were later reduced, and he was sentenced to 240 hours of community service and 12 months' supervision and ordered to pay restitution to the victim. The NFL suspended him for four games in 2016; he didn't play the entire season.