Ole Gunnar Solskjaer’s team were second in the Premier League last season but finished 12 points behind Manchester City.
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.