The injury is a major blow for the Abu Dhabi-owned club as they chase Tottenham Hotspur for fourth place in the league and a qualifying slot for next season's Champions League. City are fifth and a point behind Spurs.
Ireland international Given was given oxygen and carried off on a stretcher in the second half at the Emirates after stretching for a save and landing awkwardly.
"Shay's injury will be a problem. It's not good for us because Shay is a fantastic goalkeeper," manager Roberto Mancini told the club website.
"He is important to us, but in football you can always get injured."
"Shay is in hospital, but we have 10 days and must play three big games," he added.
Given was replaced by 23-year-old third choice goalkeeper Gunnar Nielsen, who became the first player from the Faroe Islands to play in the Premier League, because regular number two Stuart Taylor is recovering from knee surgery.
"Gunnar came on and did very well. Stuart is still some way off and everyone must be 200 percent now," said Mancini.
City also had left-back Wayne Bridge limp off but the manager said he would be okay.
"For Wayne it is not a big problem, he will be available next week," he said.
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