Gundogan has yet to renew his contract with Dortmund, which expires in June 2015.
And the 23-year-old's father Irfan - who also acts as his representative - revealed that Real are eager to take the Germany international to the Bernabeu, although he stressed that negotiations will not take place until the midfielder has recovered from a back injury that has kept Gundogan out of action since August.
"Madrid's interest in my son is considerable and they want to negotiate for him," Irfan Gundogan told AS.
"But at the moment we can't do anything until Ilkay has completely recovered from his back injury.
"When he is playing regular football we can begin talks."
Gundogan has made 84 appearances since joining Dortmund from Nuremberg in 2011, helping them to a Bundesliga and DFB-Pokal double in the 2011-12 campaign.
However, Gundogan has been restricted to just three appearances this season, although he is expected to make his return when Dortmund resume Bundesliga action on January 25.
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