The Spanish midfielder suffered the injury in Saturday's 2-1 Premier League win over West Ham at Old Trafford, and was replaced by Antonio Valencia 16 minutes from time.
United confirmed the news via their official Twitter account on Tuesday, with the club set to "monitor his progress over the next few weeks".
The injury adds to manager Louis van Gaal's injury problems, as the Dutchman now has 11 first-team squad members on the sidelines.
United host Everton on Sunday, and they will have to do so without captain Wayne Rooney, who was sent off for a wild hack at West Ham's Stewart Downing last weekend.
Herrera has made a positive start to his United career since arriving from Athletic Bilbao during the off-season, netting twice in his four appearances to date.
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