Given is seeing 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.
Asked if he would recall David Gonzalez from his loan spell at Leeds, Mancini said: "Probably. We can take David from Leeds - we don't have many more options."