Sir Alex Ferguson has publicly spoken about how he narrowly missed out on signing the French centre-back in 2011.
He has since gone on to become a star at the Bernabeu, although recent performances have brought his future into doubt.
According to the Manchester Evening News, United have a strong relationship with Varane's representatives, Base Soccer Agency.
The 27-year-old also has just 18 months remaining on his contract, which means United could avoid spending an exorbitant transfer fee.
Ole Gunnar Solskjaer wants to sign another centre-back and they now have an 'opportunity' sign a World Cup winner in Varane.