When asked about potential movement on the deal, Dortmund boss Juergen Klopp said: "I would not completely rule anything out."
The club's director of sport Michael Zorc added: "The transfer window is open until Monday."
Cleverley, also 25, is a United youth-team product but has become something of a pariah for United fans, who see him as symbolic of the club's general decline in recent years.
While there is no doubt that Kagawa is a gifted player, he has clearly not settled at Old Trafford. In his first season he impressed in fits and starts, but seemed to lose his way, apparently distracted by life in England and shunted into wide positions after Wayne Rooney and later Juan Mata were deemed superior no.10s. Last term he barely figured and completely lost confidence, although Moyes may have had something to do with that.
Dortmund, meanwhile, need to start planning for the inevitable departure of Marco Reus, whose contract expires next summer. This would give Kagawa a chance to get back into his groove - he would certainly be the fans' choice.
Cleverley, meanwhile, has become a laughing stock, through no real fault of his own, although describing himself as a Spanish style player did him no favours. His job is to keep the ball circulating in midfield, but a lack of goals and assists rankle United fans, who expect more from their midfielders.
He will do a good job for Villa and this would represent a chance to rebuild his career. If he really is as good as he believes, he could earn back his England place on merit - as Fabian Delph has.
- Sports & Recreation
- Tom Cleverley
- David Moyes
- Aston Villa
- Old Trafford