The 24-year-old picked up the problem in the latter stages of their victory over Liverpool on Saturday, but stayed on the pitch as manager Sam Allardyce had already used up all three of his substitutes.
The Senegal international went for a scan on Monday and Hammers co-chairman David Gold has revealed on Twitter that he could miss up to five games throughout October.
Gold tweeted: "Bad news. C.Kouyate will be out for 6 weeks. He will probably miss 5 games and might be available for Aston Villa at home 8 Nov.
"In the past losing a player like C.Kouyate would be devastating but now we have quality back up. Cheikhoe get well soon. dg."
Kouyate has impressed in his five appearances since joining West Ham for £7m from Anderlecht this summer.