Varane has been in and out of the side at Madrid this term and the defender has often been left out of Champions League games by manager Zinedine Zidane.
The Frenchman is said to be considering his future at the Bernabeu as a result and Man United are hoping to capitalise on the uncertainty. However, they may face competition for his signature from Bayern Munich and Paris Saint-Germain.
United were close to signing Varane, then considered to be one of the brightest young talents in the world, when he decided to leave Lens in 2011.
But a phone call from Zidane reportedly convinced Varane to make the switch to Madrid. United brought in Phil Jones instead and the rest, as they say, is history.
Daley Blind has enjoyed a solid campaign at centre-back this season but it is clear that United need to bring in a natural defender to play alongside Chris Smalling if they are to make the next step. Varane would fit the bill nicely.
The player may also be swayed by Jose Mourinho’s potential move to United, as the Portuguese is known to be an admirer of Varane, who he previously worked with at Madrid.