The German winger is currently one of the most sought after players in European football, with his future at Dortmund unclear.
Reus is understood to be stalling on signing a new deal with the Bundesliga giants, which would see a £28m release clause in his current contract activated next summer.
This has fuelled talk of offers as soon as January, with Liverpool, United and Arsenal all keen.
But reports suggest that Reus has already set his sights on a switch to Real.
The versatile attacker is believed to be keen to work under Carlo Ancelotti at the Bernabeu, where he would link up with compatriot Toni Kroos.
Reus missed the World Cup over the summer through injury and is widely considered to be one of the very best players in German football.
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