He was hurt on Tuesday in United's Champions League win over Braga.
"I think playing on with the injury did not help the situation for him, even though he did well to play on with it. It's disappointing for him," United manager Alex Ferguson said on Friday.
United has four tough games in 10 days. It plays at Chelsea twice, including a League Cup match, and then hosts Arsenal before travelling to Portugal for Braga.
Kagawa will also miss Japan's World Cup qualifier in Oman on Nov. 14.