Given will see a specialist about the problem on Wednesday and Mancini expects the Irish stopper will require surgery.
"Shay has this problem in his shoulder and I think he needs three months to recover," Mancini said. "He has the same injury that he had last year and I think that he needs an operation."
Given also damaged his shoulder in April, which prompted City to sign Marton Fulop on an emergency loan deal from Sunderland for their final three games of the 2009-10 campaign.
Mancini hopes he will not have to do something similar this time.
"I am disappointed for him because we have lost a good goalkeeper," Mancini said.
"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."
Asked if he would recall David Gonzales from his loan spell at Leeds, Mancini said: "Probably. We can take David from Leeds - we don't have many more options."