Le Parisien reported a Manchester United source claiming talks were ‘rapidly approaching’ an agreement for the 28-year-old.
The report said the main points of a deal worth around €45m (£38.5) had been agreed by the two clubs.
Ole Gunnar Solskjaer sees Raphael Varane as a dream centre-back partner for Harry Maguire
Varane could be unveiled as an Old Trafford player by the end of the week.
United had reportedly already agreed terms with the France international centre-back until 2026.
Ole Gunnar Solskjaer wants to partner Varane with captain Harry Maguire in the heart of his defence.
The World Cup winner is closing in on a sensational move, with one year left on his contract
Solskjaer looks set for a perfect transfer window, having already solved their lack of quality on the right-wing with Jadon Sancho this summer.
Their other long-standing shortcoming was a central defender to play alongside England star Maguire, with Victor Lindelof struggling to complement the £80m man’s abilities.
United are now within touching distance of providing a solution in the form of world class ability, with Goal reporting that a five-year deal has been agreed with Varane.
United boss Solskjaer is hoping Varane will be the answer to United’s issues at centre-back
The World Cup winner will receive an increased wage and is keen for the Old Trafford switch, though his respect for the Spanish champions has stopped him from agitating for the move.
Madrid have already seen Sergio Ramos exit for PSG this summer and their clear out looks set to continue with a man who has played for the club since 2011.
Carlo Ancelotti’s side are looking to recover funds following the Covid-19 shutdown, as they aim to build towards a Kylian Mbappe bid next summer.