Los Blancos are aware that Varane currently has one year left to run on his current deal in Spain and would rather command a fee for his services this summer rather than lose him on a free in 2022.
The LaLiga side are pessimistic about Varane accepting their latest contract offer, which has heightened speculation that he will leave at the end of the season.
And United, who have been linked with a move for the 27-year-old in the past, are a rumoured destination for the France international.
Varane joined Real Madrid from Lens 10 years ago, but the Spanish outfit are willing to cash in on the Frenchman as they seek to ease their financial woes.
United are looking to sign a new centre-back in the summer to partner Harry Maguire, with Ole Gunnar Solskjaer remaining unconvinced by Victor Lindelof and Eric Bailly.
The Premier League side came close to signing Varane back in 2011, but missed out on the defender after then-Real Madrid sporting director Zinedine Zidane convinced the player to swap France for Spain.
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