The German international is widely regarded as one of the best midfielders in Europe after playing a pivotal role in the Bundesliga side’s recent successes.
United are Arsenal have both been linked with January moves for his signature, but may now face major competition from Real.
The Spaniards are thought to be ready to make a £25m offer for Gundogan, as Carlo Ancelotti prepares to step up his redevelopment plans at the club following his arrival over the summer.
The La Liga side spent almost £150m before the last deadline, but are still believed to be ready to splash the cash – although the potential sale of Karim Benzema may effect their ability to open the chequebook.
Real are buoyed by the fact that Gundogan is yet to extend his contract with Dortmund.
Nuri Sahin is also on loan at the Westfalenstadion from the Madrid side, which they are hopeful can be used to their advantage.
United and Arsenal are unlikely to give up easily, with Arsene Wenger and David Moyes thought to be huge admirers of the midfielder.
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