Having already signed England international Jadon Sancho for £73m from Borussia Dortmund, the acquisition of Varane would be a further statement of intent from United.
Varane is well established as one of the world's top defenders, having joined Real from French side Lens in the summer of 2011.
He has gone on to make over 350 appearances for the club, and be capped 79 times by France, helping them to World Cup glory in 2018.
He started all four of France's matches at Euro 2020 before they were knocked out on penalties by Switzerland.
If the deal goes through as expected, Varane's first competitive action for United could come when they start their Premier League campaign on August 14 against Leeds United at Old Trafford.
'One of best central defenders in Europe'
Analysis from Spanish football expert Terry Gibson, speaking on Sky Sports News:
"If Man Utd do get Varane across the line, they are buying one of the best central defenders in Europe. He has been fantastic at Real since day one, he has all the attributes you need and he has been slightly overlooked because of Sergio Ramos.
"But in terms of quality, I do not think there are many better than Varane."