Varane has entered the final six months of his contract at Old Trafford, meaning that he is free to hold discussions with foreign clubs over a pre-contract, including Barcelona.
However, Red Devils head coach Erik ten Hag said last week that the club were in discussions with the ex-France international over a new deal.
Man United decided against activating a 12-month extension in his existing agreement, allegedly due to the fact that they want to reduce the experienced centre-back's wage packet.
There is believed to be strong Saudi Arabian interest in Varane, while Bayern Munich have also been linked with him, in addition to former club Real Madrid.
A return to Bernabeu this month is no longer an option, though, with Los Blancos head coach Carlo Ancelotti revealing that the club would not be signing a defender despite David Alaba's long-term knee problem.
According to TEAMtalk, via reports in Spain, Man United's incoming minority shareholder Sir Jim Ratcliffe is keen for the club to move Varane on this month.
Barcelona have allegedly been approached, with the Catalan outfit given the chance to sign him for €20m (£17m), with Ratcliffe wanting to remove his salary from the club, which is thought to be around £340,000 a week.
Varane has recently regained his spot in Man United's starting side, with the defender benefitting from Harry Maguire's groin problem.
Ten Hag will have a big decision to make when it comes to his starting centre-backs for the remainder of the season, though, as Maguire is due to return to action over the next couple of weeks.
Lisandro Martinez is also back in full
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