The 28-year-old centre-back has one year remaining on his Bernabeu contract but has so far shown little inclination to extend his decade-long stay with the club.
Spanish show El Partidazo de COPE report that Real would be willing to listen to offers for Varane in the current transfer window and have set an asking price.
Real Madrid 'are willing to sell defender Raphael Varane for £52million this summer' with Manchester United linked with the France international
Ole Gunnar Solskjaer and United now know Real Madrid's asking price for the Frenchman
Varane is currently with the France team ahead of Euro 2020, a tournament the reigning world champions enter as favourites.
United manager Ole Gunnar Solskjaer will target a centre-half signing to partner with captain Harry Maguire this summer as the Old Trafford club look to challenge Manchester City for the Premier League title last season.
Maguire's absence with an ankle injury in the final games of the campaign, including the UEFA Europa League final defeat to Villarreal, highlighted how the other defenders in the squad aren't up to his standard.
Signing a proven winner such as Varane would certainly remedy that. His 360 matches for Real over 10 seasons have seen four Champions League wins as well as three Spanish league titles.
Varane is currently on international duty with France ahead of the European Championship
The 28-year-old has one year remaining on his Real contract and has turned down a new one
With just one year left on his current contract and without any apparent commitment to signing a new one, it appears Varane will follow his compatriot and manager Zinedine Zidane out the exit door.
Indeed, Real would have to sell this year to ensure they get full market value for a player still in his prime and avoid losing him for nothing next year.
Varane played 41 games across all competitions for Real last season but it proved to be a disappointing campaign as they finished second in LaLiga behind Atletico Madrid and exited the Champions League to Chelsea in the semi-finals.
Varane would bring experience and a winning pedigree to Manchester United's back line
He helped France win the 2018 World Cup to add to his large collection of club silverware
A leg injury kept Varane sidelined for the second leg of that semi-final tie in an added frustration.
Other United priorities include a right-sided attacker with Borussia Dortmund's Jadon Sancho top of their shopping list.
The German club have rejected United's opening bid of £67m for the England star and are holding out for £80m.