Kicker reports that United are close to agreeing an initial £12million fee with Dortmund for Kagawa.
The deal could potentially increase to £17.5million with add-ons.
Kagawa has been a key player for Dortmund this season, scoring 17 goals for the club.
Dortmund sporting director Michael Zorc has hinted that a deal is close, stating: “Some details are still to be confirmed.”