This is according to talkSPORT, who cite reports in Italy as claiming the Red Devils are in pole position - ahead of Chelsea - to land the highly-rated France international before the 2015-16 campaign commences.
United boss Louis van Gaal will reportedly make a move for the 21-year-old at the end of the season after opting against making an approach for the defender last month.
Varane: Better than Hummels?
Van Gaal is keen to strengthen his defensive options heading into next season, when Manchester United hope to be back competing in the Champions League after a one-season absence.
The Red Devils have been strongly linked with a move for Borussia Dortmund captain Mats Hummels, but Varane may arguably be the better value-for-money signing.
This is a 21-year-old player widely considered the best young centre-back in world football, with the potential to become the standout player in his position on the planet.
Manchester United wanted Varane before he joined Real Madrid from Lens in 2011, but Jose Mourinho won the race to secure the youngster's services ahead of Sir Alex Ferguson.
Key player for Real Madrid and France
He has since established his status as a key player for both club and country, and has deservedly earned the plaudits from fans and pundits alike for his talismanic performances.
Since arriving at the Bernabeu, Varane has already won a Spanish league title, a Copa del Rey, a Champions League winners' medal, plus the UEFA Super Cup and FIFA Club World Cup.
Why would Real Madrid sell?
The French starlet may subsequently feel ready for a fresh challenge, although quite why Real Madrid would be tempted to sell him remains a mystery.
Pepe turns 32 later this month and will soon begin to decline as a central defender, while Sergio Ramos is still a top-class centre-back but turns 29 next month.
Varane is the future of Real Madrid and it would be a huge surprise if Carlo Ancelotti - or club president Florentino Perez - thought differently.
But Manchester United feel money will ultimately talk - and optimistically expect Varane could put pen-to-paper on a contract at Old Trafford this summer.