United are in the midst of a defensive crisis due to injuries to Nemanja Vidic and Phil Jones, although Chris Smalling is close to making a return after a toe injury.
There has been speculation that boss Sir Alex Ferguson could make a move for Varane, who has managed just one appearance for Real to date this season.
The 19-year-old, who joined the Spanish giants in 2011 from French side Lens, insists he has no plans to walk away from the Bernabeu and is eager to fight for a place in the Real side.
He told talkSPORT: "You could have the impression that I have not made much progress [since joining Real Madrid]. But it is false. Clearly we all have to evolve quickly, perhaps more quickly, than we might have done before. This has created some impatience, both for you and me. But for the moment I do not have reason to think about a transfer."
Ferguson has also been linked with a January move for Fulham's Brede Hangeland and it would seem as if the United boss is keen to bolster his defensive options in the winter transfer window, even if he cannot convince Varane to quit Madrid.