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.
Former boss Sir Alex Ferguson originally pursued a deal for the player before his move to Madrid, and Jose Mourinho was also keen to add the defender to his squad during his time at Old Trafford.