The Red Devils are thought to be desperate to get the German World Cup winner to Old Trafford this summer, with manager Louis van Gaal keen on the commanding enforcer.
With Nemanja Vidic and Rio Ferdinand having been released, United are short on experience at the back, which is seen as a problem by the Dutchman as he looks to usher in a new three-man central defensive unit.
Hummels is seen as the ideal player to step in, but Dortmund are, understandably, reluctant to let him leave.
But the 2012/13 Premier League champions are ready to tempt them into letting him go by offering Kagawa as bait.
The Japanese playmaker joined United from the Bundesliga side in 2012, and has been heavily linked with a return to his old stomping ground after failing to make an impact at the Theatre of Dreams.
BVB are thought to be keen on him, while fans of the club have even made online petitions to get him back to the Westfalenstadion in the past.
With LVG having suggested that his inherited squad is a little top heavy, he may well be eager to get rid of the Asian star if it means addressing pressing concerns.
United are also ready to make a similar offer for Thomas Vermaelen, with Arsenal rumoured to be watching Nani.