Kiffin had previously said Barkley, who has been nursing a sprained right shoulder for more than a month, would be re-evaluated when the Trojans arrived in El Paso, Texas for the game.
Barkley tweeted on Thursday that he was not medically cleared to play in the game. He has not made a decision on whether or not to play in the Jan. 26 Senior Bowl, which could have a sizable impact on his stock for April's NFL draft.
"Doctors haven't cleared me to play, so I won't be suiting up on Saturday. I'll be here with my team to finish strong. Proud to be a Trojan!" Barkley tweeted.
A senior and one-time Heisman Trophy contender, Barkley has been sidelined since suffering the injury late in the fourth quarter of a Nov. 17 USC loss to rival UCLA.
Wittek, a 19-year-old redshirt freshman, made his first career start in USC's next game, a regular season-ending loss to Notre Dame, and completed 14 of 23 passes for 186 yards, one touchdown and two interceptions.
Wittek has been preparing to start the Sun Bowl for the last two weeks, at first splitting first-team quarterback reps with fellow freshman Cody Kessler and then taking the majority of late.