The France international, who has made a big impact for the London club since signing from Marseille this year, was injured by a tackle from Everton's James McCarthy and hobbled off early in the second half.
"(Payet) suffered an injury to his left ankle during Saturday's game with Everton," West Ham said on their website (www.whufc.com).
"He had a scan on Sunday and after a consultation with a specialist on Monday, it is likely that he will be ruled out of action for three months."
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Payet's absence is a blow for sixth-placed West Ham as he has scored five Premier League goals this season.
West Ham forward Enner Valencia also injured his ankle against Everton and will be assessed over the next few days.
(Reporting by Martyn Herman, editing by Ed Osmond)
