The Japanese international was substituted at halftime of United's 3-2 victory over Braga in the Champions League on Tuesday.
"Shinji Kagawa is going to be out for three to four weeks," Ferguson told reporters.
"It's a knee injury similar to the one Ashley Young had but not as bad.
"I think playing on with the injury did not help the situation for him, even though he did well to play on with it. It's disappointing for him."
The injury means Kagawa will be unavailable for Japan's World Cup qualifier away at Oman on November 14.
Meanwhile, defenders Chris Smalling and Phil Jones are nearing a first team return after missing the start of the season through injury.
"Smalling is back training on the football side of things," Ferguson added.
"Phil Jones will hopefully start full training a week on Monday, so things are getting better on that front.
"Certainly getting Smalling back is a big bonus for us. Other than that, everyone else is available for Sunday's game [at Chelsea]."
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