The 29-year-old suffered the injurying during the 1-0 defeat by Tottenham on Wednesday.
The injury could affect Barry’s chances of being named in England boss Fabio Capello’s provisional 30-man squad for the World Cup.
“I feel very sorry for him,” said City manager Roberto Mancini.
“Gareth is a fantastic guy and a fantastic player.
“I don’t know exactly how bad, maybe three to four weeks. I hope he can play in the World Cup but Fabio Capello and his medical staff must decide.
“He deserves to play in the World Cup”
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