Varane had been a longstanding target for United, who last week completed the transfer of England winger Jadon Sancho from German club Borussia Dortmund for a reported £73 million ($100 million).
Ole Gunnar Solskjaer's men are now closing in on the signing of Real centre-back Varane, having agreed a deal worth a reported £34million rising to £42million with add-ons.
The 28-year-old Varane, a 2018 World Cup winner, has one season left on his contract at the Bernabeu.
He moved from French club Lens to Madrid in 2011 and has since made 360 appearances across all competitions for Real, winning 18 major honours -- including four Champions League titles.
Should his move to Old Trafford be completed, Varane would find himself vying for a place in the centre of United's defence with club captain Harry Maguire, Eric Bailly and Victor Lindelof.
United finished second in last season's Premier League but were a huge 12 points adrift of title-winners and local rivals Manchester City in the race to be crowned champions of England.
The Red Devils have not won a major trophy since the 2017 Europa League, with United's last Premier League title success coming in celebrated manager Alex Ferguson's final season before retirement eight years ago.
