Vidic, who missed the final five months of last season after rupturing his cruciate in the Champions League defeat to Basel, missed Sunday's Premier League win at Liverpool after complaining of tightness in his right knee.
However, a subsequent examination revealed a greater problem, prompting Vidic to be sent for an operation, which has now been completed.
"Nemanja Vidic has had a meniscus operation on his right knee and will be out for around eight weeks," said a club spokesman.
"Sir Alex, as a precautionary measure, rested Vidic at the weekend, after he complained of tightness in his knee, however, further medical investigation revealed the underlying problem.
"The central defender was operated on this week and will be sidelined for about eight weeks."