It is understood the clubs have agreed an initial fee of £34m, which could rise to £42m, and that Varane, 28, has agreed terms on a four-year contract with the option for another 12 months.
United said the transfer would be completed subject to a medical and to player terms being finalised.
Varane, a World Cup winner with France, has made 360 appearances for Real.
Real Madrid wants to show all its affection to Raphaal Varane and his family and wishes him good luck in this new stage of his professional career, the club said in a statement.
Our club thanks you for your professionalism and exemplary behaviour during the 10 seasons in which you defended our shirt.
Varane, who has 79 caps, will become Ole Gunnar Solskjaer’s third signing in this transfer window – following the re-signing of goalkeeper Tom Heaton and the £73m arrival of England winger Jadon Sancho from Borussia Dortmund.
BBC Sport understands Varane will quarantine before having his medical, after which he will complete his move.
United, who agreed a fee with Real on Monday, have been long-term admirers of Varane, whose contract was due to expire next summer.
After coming through Lens’ academy, Varane broke into the first team at the age of 17, and while he was unable to steer them away from relegation, his performances attracted the attention of some of Europe’s top clubs.
Despite strong interest from United, Real signed the teenager on a six-year contract in 2011, and he went on to make 15 appearances in all competitions during his first season in Spain as Mourinho’s side regained the title.
That was the start of Varane’s impressive trophy haul during his decade in the Spanish capital.
He helped Carlo Ancelotti’s side clinch their 10th European title – ‘La Decima’ – in the 2013-14 season, making seven appearances and partnering Sergio Ramos in the final as they beat Atletico Madrid 4-1 after extra time in Lisbon.