The Daily Mail is reporting that the Red Devils are long-term admirers of the France international since Sir Alex Ferguson missed out on the signing of Varane back in 2011.
The same article states that Manchester United are looking at the French defender as a potential centre-half partner for club captain Harry Maguire ahead of the 2021-22 campaign.
According to the same story, Varane could be set to leave Real Madrid this summer when the Frenchman will only have 12 months left to run on his current deal.
The report goes on to state that Real Madrid will be looking to secure a transfer fee in the region of £60m for the France international as los Blancos look to raise funds.