The Dutch striker scored a hat trick against Olympiakos in the Champions League on Wednesday night but was stretchered off late in the game and was revealed to have suffered a knee problem.
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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 seen to 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 quarterfinal tie with last year's winner Bayern Munich, which was 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.