Bush has 809 carries for 3,250 yards and 29 touchdowns along with 104 receptions for 1,010 yards and two touchdowns over a seven-year NFL career.
Bush entered the league as a fourth-round pick of the Oakland Raiders in the 2007 draft. He sat out his rookie season because of a leg injury then played for the Raiders from 2008 to 2011 before signing with the Chicago Bears in 2012.
He played in 15 games with the Chicago Bears last season before being released in March.
The Cardinals also released tackle Kelvin Palmer from the practice squad Tuesday.