Varane is already understood to have agreed personal terms with the Red Devils, but the progress of the deal has stalled during talks over a transfer fee.
Man United were keen to drive down the price from Real Madrid's £42.8m mark, but Marca reports that they are close to giving in to Los Blancos' demands.
Varane's contract in the Spanish capital expires next summer and Madrid are eager to cash in on him this year rather than risk losing him on a free transfer.
Madrid have already let Sergio Ramos leave the club for free this summer, a player whom Varane played alongside for a decade at the Bernabeu.
The 28-year-old has made 360 appearances for the club since joining from Lens in 2011, helping them to three league titles and four Champions League crowns in that time. body check tags ::