The Frenchman has become one of the world’s best defenders during his time in the Spanish capital, winning four Champions League titles with Los Blancos.
But he is entering the final year of his contract at the Bernabeu and looks likely to depart Madrid in this summer’s transfer window.
The Red Devils have been the frontrunners to sign Varane for much of the summer, but there is confusion over whether they have yet submitted a bid for the 28-year-old.
United are searching for a new partner for Harry Maguire at the heart of their defence, with Eric Bailly and Victor Lindelof having failed to be consistent performers.
Varane is their top target to play alongside Maguire at centre-back as Ole Gunnar Solskjaer looks to mount a Premier League title challenge next season.
The Red Devils are set to make Jadon Sancho their first signing of the summer having agreed a fee with Borussia Dortmund.
They are now turning their full attention to Varane, though Madrid’s price tag for the defender has not yet been stated.
According to The Athletic, Varane’s representatives have told Madrid that their client wants a new challenge having spent a decade in Spain.
He has not yet signed an extension to his contract and Madrid president Florentino Perez will need to sell him this summer if they are to bring in a substantial fee for him.
They are also said to be “urgently” in need of income due to the impact of the Coronavirus pandemic, with Varane one of their top assets.
But United’s hopes of completing a deal will hit by the blow that The Athletic state that Madrid are holding out for “significantly” more than United’s valuation.
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The theory that the Red Devils are now relying on though is that as the window edges closer to ending, Madrid may have to slash their asking price.
That could mean Varane is available for a fee in the region of £40m-£45m, though United’s valuation is not yet known.
It is claimed that sources close to Madrid believe that Varane will indeed move to Old Trafford, with personal terms likely to be swiftly agreed.
But United are facing competition from Paris Saint-Germain for the signing of Varane, with the French club having already added his former centre-back partner Sergio Ramos to their squad.
The Athletic state that PSG are indeed interested in Varane, though the signing of Ramos does leave little room for him in Mauricio Pochettino’s squad.
They are also unlikely to want to be drawn into a transfer battle with Madrid as they desperately try to convince Kylian Mbappe to stay at the Parc des Princes, rather than move to the Bernabeu.
That leaves United as the only likely contender to sign Varane, but they will have to make a move soon to allow him to settle in Solskjaer’s squad.