The Japan international has been a long-term target for United manager Sir Alex Ferguson.
Kagawa is inside the final two years of his current deal.
He said at the end of last season: "If I get a formal offer, I'll be strongly interested in listening to what they have to say.
"The Premiership is one of the leagues I want to challenge. If it is (United), I'll be still more eager to go and try and see how much I can do there."