The 29-year-old was forced off after aggravating the long-standing injury in the first half of City's 1-1 draw at Sunderland on Sunday and will now miss a crucial phase of the season, with the Eastlands club bidding to secure fourth place in the Premier League and a place in the Champions League.
Bridge had only recently returned to first-team action for Roberto Mancini's side.
His comeback in the loss at Hull on February 6 was played out in the glare of an intense media spotlight following allegations of an affair between his former partner Vanessa Perroncel and ex-Chelsea team-mate John Terry.
Bridge subsequently brought an end to his World Cup chances by announcing he would not accept an invitation from Fabio Capello to play alongside Terry in the England set-up.