The 23-year-old played in Japan's 1-0 win over UAE on Friday but was named on the bench by his national side's boss, Alberto Zaccheroni, for the match against Iraq.
Kagawa played the first half against UAE but was substituted at half-time after Zaccheroni was reportedly frustrated with one of his star players' displays.
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Kagawa himself has played down fears that he could be missing for some time for the Premier League champions, stating that he should be back soon.
"It [the injury] happened in yesterday's practice," he told Nikkan Sports. "I was holding the ball and went to make a move and in that instant it started to hurt.
"I won't take long [to recover]."
Manchester United host Wigan on the Premier League on Saturday before kicking off their Champions League campaign on the following Wednesday with another home match against Turkish side Galatasaray.