The Dutch striker scored a hat-trick against Olympiakos in the Champions League on Wednesday night but was stretchered off late in the game, having suffered a knee problem during the closing stages of the encounter.
United boss David Moyes initially declared that the injury "did not look too bad" but later revealed that the club would send the 30-year-old for scans on Friday.
After Van Persie had been examined by the club's medical team, United confirmed that the striker could miss up to six weeks with the knee injury.
Crucially, the problem rules Van Persie out of Manchester United's Premier League clash with Manchester City, as well as both legs of United's Champions League quarter-final tie with last year's winners Bayern Munich, who were paired with Moyes' side on Friday.
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Van Persie, however, is likely to be fit for the World Cup, where he will represent the Netherlands in Brazil.