The 17-year-old Lens defender is touted as a rising star in France.
United are ready to put down £3.7million for the player.
Reports have also linked Arsenal and Chelsea with a move for Varane.
Highly rated: Varane (below) in action for Lens against Brest back in November
Meanwhile Tomasz Kuszczak believes he is good enough to replace Edwin van der Sar on a long-term basis and claims there is no need for Manchester United to find another top goalkeeper.
Van der Sar, currently suffering from flu, is debating whether to retire from football at the end of the season.
United have continued to be linked with several goalkeepers despite signing Anders Lindegaard from Aalesund.
But Kuszczak is determined to become the number one, or move on.
The 28-year-old told the Manchester Evening News: 'I know I have not got the biggest name in the world, and I do not want this to sound arrogant. I think I am good enough.
'I have talked to the boss a few times and he said we will see what happens at the end of the season.
A tale of two keepers: Tomasz Kuszczak (left) believes he can replace Edwin Van der Sar (right)
'I know that in the meantime if I get a chance I have to play well because that is the only way I am going to impress him.
'If Sir Alex Ferguson decides I am not good enough, then I will look for another club.
'Everything is open now. I am realistic about my situation. It would not frighten me to take over as Edwin's successor. Why should it? I am not new here.
'If I play for this club long term I will be really happy because that has always been my dream. I came to Manchester to try to be a top goalkeeper for many years like my idol Peter Schmeichel.'
City and United on alert as unsettled Santos star Ganso hits out at club Valencia to miss Man United's Champions League tie in MarseilleMANCHESTER UNITED FC