The England striker picked up the knock while on international duty but is expected to recover in time for next Monday's Premier League meeting with Manchester City
The England striker was also left out of the Red Devils' 1-0 victory over Sunderland at the weekend, with the expectation that he was being rested ahead of the replay at Stamford Bridge.
Javier Hernandez and Danny Welbeck instead get the nod to feature in attack against the Blues although Sir Alex has revealed that a groin complaint, picked up on international duty with England, is the reason behind his absence.
The Scot told ITV: "He got a groin injury playing for England. He wasn't 100 per cent yesterday in training so there's no point risking him. If you exacerbate a groin injury it can put you out for weeks."
United face Rafael Benitez's men looking to book their place in the semi-finals at Wembley, where fierce rivals Manchester City already await the victors of the replay.