The Senegal international sustained the injury during the second half of the 3-1 victory at Upton Park on Saturday, but stayed on to complete the 90 minutes after receiving treatment.
On Monday, West Ham's co-owner David Gold announced the news via his official Twitter account, with Kouyate set to be out until November.
"Bad news. C.Kouyate will be out for 6 weeks," he posted.
"He will probably miss 5 games and might be available for Aston Villa at home 8 nov [November]. dg.
"In the past losing a player like C.Kouyate would be devastating but now we have quality back up. Cheikhoe [sic] get well soon. dg."
Sam Allardyce's side travel to Manchester United on Saturday.
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