"The France international came off at halftime in our 3-2 win against Nottingham Forest at Old Trafford on Saturday due to the complaint, which is expected to keep him out for a few weeks," United said in a statement.
Victor Lindelof (29) replaced Varane on Saturday and manager Erik ten Hag (53) said it was for injury reasons.
"He had complaints, we didn't want to risk him, we have Harry Maguire (30), Victor Lindelof, who could replace him, so we don't want take a risk," Ten Hag said after the game.
"He had complaints and if it's really bad I don't know, we have to wait."
The extent of the injury has not yet been revealed and Ten Hag is expected to provide a further update on Friday.
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