Ole Gunnar Solskjaer is keen to sign a centre-back this summer and has identified the Frenchman as his primary target.
According to the Manchester Evening News, there is optimism amongst the Red Devils camp that a deal for the 28-year-old could be completed by the end of this month, before the beginning of the new Premier League season.
The transfer fee is not expected to be a stumbling block between the two clubs, with United expected to pay around £50m.
Varane is currently on holiday with his family following his participation with France at Euro 2020, though if a deal to join United is completed, then he would intend to fly straight to Manchester instead of returning to Madrid.
Real are looking to sell Varane this summer as he has just one year remaining on his contract, and there have been no indications to suggest the defender wishes to extend his deal with the Spanish club.
Varane has spent 10 years at the Santiago Bernabeu, recording 360 appearances and winning 18 major trophies, including four Champions League titles. body check tags ::