The Frenchman is approaching the final year of his contract in July and has yet to be convinced by the Madrid giants to commit his future to the club.
With the lack of progress in contract talks, Varane could be sold by Los Blancos in a few months. United and Les Parisiens are said to be ‘very interested’.
However, it is reported that the Blues are favourites to meet the €70 million valuation for the centre-back at the ‘express request’ of manager Thomas Tuchel.
Varane has been a brilliant servant for Los Blancos over the past decade, registering 357 appearances across all competitions.
He has established himself as a world-class defender and remains an undisputed starter for them.
Despite this, he appears to be mulling over a fresh challenge, and it has been reported that he would prefer a Premier League move.
United failed to sign him from Lens back in 2011 and were unsuccessful with another attempt to lure him from Los Blancos two years ago.
They could return for his services in the forthcoming transfer window but may be reluctant to go on a bidding battle with their league rivals.
The Blues spent more than £225m on signings last year, and they are likely to back Tuchel with transfer funds.
The Blues have a big decision to make on the futures of Cesar Azpilicueta, Andreas Christensen and Antonio Rudiger, whose contracts expire in 14 months.
Thiago Silva’s deal is due to conclude at the end of the campaign, but the 36-year-old appears most likely to sign a 12-month extension.
Stats from Transfermarkt.com