It had been reported on Friday morning that the England international could be out of action for three months with glandular fever but Sir Alex set the record straight at his press conference ahead of the game with struggling Rovers.
The cultured defender should be fit to play at Newcastle in United's first assignment of 2012 but is one of several stars sidelined for the New Year's Eve encounter at Old Trafford.
"Chris Smalling has got tonsillitis," confirmed Sir Alex. "I know there are reports about glandular fever but they're off the mark.
"He had glandular fever as a kid but that's not the case here. He should be ready for the Newcastle game, at least definitely the Manchester City game in the FA Cup. He'll be ready for those games."
The other injuries are causing some frustration at Carrington after problems at the back forced a major reshuffle against Wigan.
"We don't have much change," updated the boss. "Obviously injuries to the defenders are quite a concern now, with Jonny Evans taken off at half-time [on Monday]. We're hoping Phil Jones will be fit, there's possibly an outside chance that Rio Ferdinand will get on the bench. Other than that, we're the same."