The Real president Florentino Perez is reportedly considering accepting the bid as he wants to partner Barcelona’s Samuel Umtiti alongside Sergio Ramos next season.
Varane joined Real from RC Lens in 2011 and has gone on to make 229 appearances in all competitions for Los Blancos.
He’s been a regular at the heart of the defence this season, making 40 appearances in La Liga, the Champions League, Copa Del Rey, Supercopa, UEFA Supercup and Club World Cup, but has an uncertain future this summer.
United have been desperate for a world-class centre-back ahead of the new campaign, as summer flop Victor Lindelof has been linked with a loan move away from Old Trafford, while Daley Blind and Marcos Rojo are expected to be sold when the transfer window opens.
Umtiti was on the Red Devils’ shortlist, but the French international has more chance of replacing countryman Varane at Real instead.
West Bromwich Albion’s Jonny Evans was also named as a potential option, having spent more than a decade at Old Trafford previously. He’d cost just £3m.
Chris Smalling and Phil Jones have formed a solid partnership for United in recent weeks, but a stellar defender like Varane could turn them into genuine title challengers.