The Dutch international is to miss the Premier League derby at home to title-chasing Manchester City on Tuesday and next month's Champions League quarter-final against holders Bayern Munich.
"Robin van Persie has a sprained knee which will keep him out for around four to six weeks," United said on their website (www.manutd.com).
The 30-year-old was injured in the 90th minute of the 3-0 victory over Olympiakos at Old Trafford on Wednesday.
He left the field on a stretcher having scored a hat-trick to give United a 3-2 aggregate win.
The Premier League champions, in seventh place in the table, visit West Ham United on Saturday.
(Editing by Ken Ferris)
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