The Los Blancos have reportedly set a €70 million asking price for the wantaway centre-back who is currently a prime target for the Red Devils.
The four-time Champions League winner has recently been linked with a move to Old Trafford which could materialise in the coming weeks.
But Real Madrid seem to be reluctant to offload their highly-rated defender after losing experienced Sergio Ramos, who is set to join French giants Paris Saint-Germain, on a free transfer.
United are desperate to bring in Varane to beef up their defensive department and have made a €40 million-plus-bonuses offer to Real Madrid for the former Lens defender, according to Sky Italia.
Varane is reportedly keen on a move to the Premier League as he's yet to show a desire to extend his stay at Real Madrid, having just one year left on his current contract.
Ole Gunnar Solskjaer has been a fan of Varane for sometime and sees the inclusion of the defender to be a perfect signing for his side after completing the signing of Jadon Sancho from Dortmund.
The the potential signing of Varane is likely to help Ole Gunnar Solskjaer's side close the gap on Pep Guardiola's Manchester City side after finishing second behind the Citizens last season.
Manchester United will be hosting Leeds Unted on August 14 at Old Trafford.
Varane has since earned 79 caps for the France national team, starting all seven games in their triumphant World Cup in 2018.
After United lost the Europa League title to Spanish outfit Villarreal, several Man United legends including Gary Neville have opted for defensive reinforcement.
Varane recorded 31 appearances with 2 goals for Real Madrid in the Spanish La Liga during the 2020/21 season when the White Angels finished second behind champions Atletico Madrid.
Having joined Real Madrid from Lens in 2011, Varane signed a six-year contract with a transfer fee believed to be in the range of €10 million.
