The Frenchman, who joined Madrid over United in 2011 after an intervention from Zinedine Zidane, is believed to be keen on leaving the Bernabeu to seek a new challenge.
Uniteds board vetoed moves for Jerome Boateng and Diego Godin last summer against the wishes of Jose Mourinho as the club wanted to pursue a long-term solution to their problems in central defence.
The club briefed last summer that they would be willing to surpass the £75million Liverpool spent on Virgil van Dijk for the right centre-back and Varane is considered the perfect choice.
Zinedine Zidane has insisted publicly that he wants to keep Varane but Real Madrid are set for a summer overhaul and need cash to fund moves for Eden Hazard and one of Kylian Mbappe or Neymar.
Thats opened up an opportunity for United and the Independent claim that the Red Devils have initiated contact with Varane to discuss a move.
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The Frenchman is likely to cost in excess of £100m – a club record fee that would surpass the £89m paid for Paul Pogba.