The 23-year-old sustained the problem while on international duty with Japan over the weekend.
Despite originally playing down the severity of the injury, Kagawa has now confessed that "it still hurts" when he puts any strain on the area.
"I've had treatment for as long as possible. I can walk without any problem but if I put strain on it it still hurts," he told reporters.
As a result Kagawa has been rated doubtful to feature during United's home encounter against Wigan Athletic on Saturday.