The 24-year-old is currently with the Japan squad ahead of international friendlies against Guatemala and Ghana.
Kagawa has been overlooked by new United manager David Moyes, and his last first-team match for the Red Devils came in the FA Community Shield at Wembley when he came on for the final seven minutes as a late subsititute.
Kagawa said: "Tough times are continuing, but all I can do is to work hard."
Asked whether he would be staying at Old Trafford, he added: "Yes, that's right".
Fans of his former club Borussia Dortmund have been using Twitter to urge the club to re-sign Kagawa.