Ayew sustained the injury in his country's 2-1 defeat to Egypt in the second leg of their World Cup playoff on Tuesday.
Ghana qualified for the 2014 tournament with a 7-3 aggregate victory.
Ayew, who will undergo surgery on Monday, will miss Tuesday's Champions League trip to Arsenal and is not expected to return to action until after the European winter break.
Marseille are already out of contention in the Champions League and lie fifth in Ligue 1, 10 points adrift of leaders Paris St Germain.
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