United are desperate for a new centre-back and have been heavily linked with the France star, 24, in the past.
Further reports that emerged last week claimed United were previously offered the World Cup-winning centre-back but they baulked at the £70million asking price.
Real are said to be demanding a similar fee before they enter negotiations this summer.
But United’s stance in the fee could now change as Express Sport understands manager Ole Gunnar Solskjaer will be given significant funds to strengthen his squad for next season.
Norwegian Solskjaer has overseen a dramatic upturn in form since taking over in the interim from Jose Mourinho last December and was rewarded with a permanent three-year contract last week.
After winning 15 of his 20 games in charge, the former Red Devils striker will now be given a big say in what players are brought into the club.
Signing a new centre-half remains a priority and, as well as Varane, they have been linked with Kalidou Koulibaly of Napoli and Leicester’s Harry Maguire.
United are also tracking West Ham star Declan Rice and Borussia Dortmund sensation Jadon Sancho.
Solskjaer admits he is already working on some potential new signings but insists he will only bring in those who can fit in with the club’s philosophy.
He also sent out a warning to the players already at Old Trafford by saying he would get rid of anyone who becomes complement under his watch.
“The players who get complacent, they never last at this club anyway,” he said before Saturday’s 2-1 victory over Watford.
“I’m the same. I can’t be complacent and rest, now I’ve got the job.
“I’ve got some targets. I want a Manchester United team that’s one of the hardest-working and fittest teams in the league.
“That then will bring results. The players know my expectations on the future.”