"The nation can breathe again. We're relieved, it could have been worse," United manager Alex Ferguson told reporters.
Rooney, who has been in superb form this season scoring 34 goals for the league leaders, sustained the injury in Tuesday's 2-1 Champions League quarter-final first leg defeat by Bayern Munich.
Pictures of him limping off at the Allianz Arena had triggered fears that the 24-year-old could miss England's World Cup campaign in South Africa.
His absence comes as a big blow to United, who host second-placed Chelsea on Saturday with just a point separating the two title rivals after 32 games.
Rooney is set to miss Wednesday's Champions League second leg at home to Bayern and, if United progress, possibly the semi-final first leg against Olympique Lyon or Girondins Bordeaux on April 21.
With United gunning for a record 19th league title, Rooney could also sit out away games at Blackburn Rovers and rivals Manchester City, meaning Ferguson is likely to hand chances to fellow strikers Dimitar Berbatov and youngster Federico Macheda.
"(Macheda's) one of the most exceptional young strikers I've seen in my time at United," Ferguson told the club website.
"He has every chance of taking part in big matches. His ability around the box for a young player is outstanding."
Ferguson said long-term injury absentees John O'Shea and Owen Hargreaves have been training well and will now be considered for substitutes places against Chelsea and Bayern.
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