The Japan international sustained the injury during the Champions League victory over Braga in midweek.
Sir Alex Ferguson said in his weekly press briefing on Friday that Kagawa will be absent for around "three or four weeks", starting with Sunday's Premier League fixture against title rivals Chelsea.
"It's a knee injury similar to the one Ashley Young had," Ferguson said. "I think playing on with the injury did not help the situation. It's disappointing for him."
Though goalkeeper Anders Lindegaard remains sidelined with a thumb injury, United did have more positive news on the recoveries of defenders Chris Smalling and Phil Jones.
"Smalling is back training on the football side of things," added Sir Alex. "Phil Jones will hopefully start football training a week on Monday.
"It's a big bonus to have Smalling back and everyone else is fit."