Given picked up the injury in the warm-up for the Europa League clash with Aris Salonika last week.
City boss Roberto Mancini said: “Shay has this problem in his shoulder and I think he needs three months to recover.
“He has the same injury that he had last year and I think that he needs an operation.
“I am disappointed for him because we have lost a good goalkeeper.
“Last year we had the problem in the last three games because we lost him to injury and I hope this year it is not the same.”