The Frenchman came off at half-time during a dismal 1-1 draw with Brentford on Saturday.
The Red Devils were unable to keep pace with Aston Villa and Tottenham in the top four race.
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Varane, who is 31, was asked if he would be available for the coming games.
He told the Manchester Evening News that he was not too bad" with respect to his injury.
Manager Erik ten Hag added: "I can't tell by now. Rapha we took him off as a precaution at half-time and Victor (Lindelof), he came off but we have to assess him tomorrow."