Foster was taken off in the 77th minute of Saturday's 0-0 draw at Everton.
The former Manchester United goalkeeper, who started both of his side's Premier League games this season, will undergo surgery in the next few days.
Steve Clarke is now left with Boaz Myhill and Luke Daniels, who replaced Foster on Saturday, as replacements.
Clarke told the club's official website: "There’s no hiding the fact this is disappointing news as Ben is a key player for us. He was excellent on Saturday before his injury forced him off.
"In the meantime, I’ve got two other very good goalkeepers to choose from.
"Boaz did well when he deputised for Ben last season and Luke showed on Saturday he is a more than capable goalkeeper."
Daniels made his debut for the club when he came on for Foster at Goodison Park and the 25-year-old may make his first start against Newport County in the League Cup on Tuesday.
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