The French international has previously been on United's radar, having held talks with the club back in 2011, before opting for a move to the Bernabeu.
That led to United turning to then-Blackburn defender Phil Jones, but the club have maintained an interest in Varane since and manager Jose Mourinho is also thought to be a big admirer, having previously worked with the former Lens man during his time at Real Madrid.
According to English publication 'The Mirror', United sources have hinted at a fresh move for the 24-year-old, who only signed a new five-year deal last September.
The same source claims that 'Los Blancos' would be open to a deal, but that they would insists on a fee in the region of £50m, some £10m more than United are reportedly wiling to pay.
Mourinho is thought to be planning a squad shake-up in the summer, with the likes of Luke Shaw, Chris Smalling, Victor Lindelof and Phil Jones all seen as replaceable by the Portuguese.
However, it would be a surprise if Real opted to let Varane go, with the Frenchman an integral part of Zinedine Zidane's starting lineup when fit.