The Daily Mail reports that United manager David Moyes has set his sights on the German, as well as his teammate Ilkay Gundogan, who also looks likely to reject a move to Old Trafford.
The article suggests that Dortmund are lining up a new deal with the 24-year-old in order to remove a £30million release clause that could see him snatched away from the Bundesliga and Champions League runners-up.
Gundogan, meanwhile, is thought to have chosen Real Madrid as his next team with the Spanish giants prepared to throw £25million at a possible deal, the Daily Mail reports.
Moyes has been knocked back in several attempts to sign a midfielder since taking over from Sir Alex Ferguson as manager.
Reports have suggested that he will try to use former Dortmund player Shinji Kagawa as part of a swap deal with one of the club’s talented midfielders.