Rosen was injured in UCLA's 23-20 loss to Arizona State on Oct. 8 and has missed the team's past three games. He did not travel with the team to Colorado for Thursday's game, a 20-10 victory for the 15th-ranked Buffaloes.
The specifics of the injury are unclear. UCLA coach Jim Mora indicated last month that Rosen was having an issue with a nerve in his shoulder.
"What happens is the nerve has to start firing again and once it starts firing again he should be fine," Mora said. "It's just a matter of when it starts firing again. He's getting all the treatment he can get."
Earlier this week, he clarified that there is no actual nerve damage and a UCLA spokesman said the same Thursday. The quarterback is not scheduled to have surgery, the spokesman said. A timetable for his recovery was not provided.
"We're going to always do what's best for our players and their long-term care, their long-term health, their long-term well-being," Mora said Tuesday. "Sometimes that means protecting them from themselves."
After a brilliant freshman year, Rosen entered the season as a Heisman Trophy candidate, but the offense struggled after changing to more of a pro-style system. In six games, Rosen completed 137 of 231 passes for 1,915 yards with 10 touchdowns and five interceptions. Redshirt senior Mike Fafaul, a former walk-on who started the past three games, is expected to be the starter for the rest of the season.