Coach Nick Saban said Monday that both will have to undergo knee surgery after injuries sustained during the Mississippi game.
White has eight catches for 105 yards and two touchdowns, including a 51-yarder against Michigan in the opener. He was hurt on the Crimson Tide's first drive.
Hart was injured after fielding a fourth-quarter punt. He was a top recruit in 2011 but was sidelined last season after a knee injury during the summer. That leaves Eddie Lacy and freshmen T.J. Yeldon and Kenyan Drake as the only scholarship backs.
Receiver Chris Black, running back Jalston Fowler and reserve safety Jarrick Williams are also expected to be out for the season with injuries.