Bush, who specialized in carrying the ball in high-pressure situations near the goal line, scored five touchdowns for Chicago this season before succumbing to a rib injury.
He ran for 411 yards on 114 carries over 13 games, while also catching nine passes for 83 yards.
His roster spot was filled when the Bears re-signed running back Kahlil Bell, who returns to the team after being released earlier this season.
Bell, who has gained 589 rushing yards on 131 carries over a four-year career, was cut by New York Jets this week.
Swim Week Miami Beach 2014 [PHOTOS]