According to the Manchester Evening News, Varane is a longstanding target for the club and is seen as an ideal partner for Harry Maguire.
While Real are open to selling the centre-back, who has one year left on his contract, they are believed to be looking for a fee nearer to £80million.
United will hope to reach a compromise considering Varane will be able to move for free next summer and president Florentino Perez has acknowledged a need to address the club’s mounting debts.
Their £50million bid is seen as a starting point for negotiations, which are expected to continue over the coming weeks.
After an excellent showing in France’s opening group game against Germany, Real believe that Varane’s price could rise if he keeps impressing at the Euros.
As well as winning a host of honours at club level, including the Champions League four times, Varane starred in his country’s success at the 2018 World Cup.
With France one of the favourites to win the tournament, he will be looking to add a European Championship medal to his impressive collection from a decade at the Bernabeu.
The 28-year-old joined Real from Lens as a teenager for just £10million, before blossoming into one of the world’s best centre-backs.
Known for his physical dominance, excellent reading of the game and composure on the ball, he is seen as the man to bring the best out of Maguire.
Man United’s captain has typically played alongside Victor Lindelof over the last couple of years, but Ole Gunnar Solskjaer is known to be in the market for an upgrade.
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