Varane has a muscular injury that could mean the end of his season for the Premier League club.
Given he is out of contract in the summer, Varane is racing to ensure he has not already played his last game for United.
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United's statement read: "Manchester United defender Raphael Varane will be out for the next few weeks with a muscle injury sustained during our game against Chelsea last week.
"The French World Cup winner is aiming to be back in action before the end of the season in May."
United added that veteran defender Jonny Evans will also be out of action with a muscle problem.