Manchester United and Juventus are the two clubs interested in the French defender who is said to be open to a move away from the Bernabeu.
But Real Madrid’s €100m (£86m) asking price could scupper any possible move from the two clubs.
United are in desperate need of a dominant centre-back and Varane fits that profile.
They have previously enquired about the World Cup winner but have never been able to get close to a deal.
Italian news outlet Calciomercato revealed that Los Blancos are keen to tie Varane down to a new contract and a hefty asking price should allow them time to negotiate with the 26-year-old.
They have also reported that another unnamed Premier League club have contacted the Spanish side over Varane’s availability.
Los Blancos president Florentino Perez is a big fan of the French defender and doesn’t see any reason to sell, especially with Zinedine Zidane looking to rebuild the squad this summer.
Their interest in Varane follows other transfer news:
Former United boss Jose Mourinho was very vocal over his need for a new centre-back before he was dismissed earlier this season.
The Portuguese manager recruited Eric Bailly and Victor Lindelof during his time at the Premier League club, but neither have flourished like he hoped.
He attempted moves for Diego Godin and Jerome Boateng, along with Varane, during his time at Old Trafford.
Chris Smalling and Phil Jones are the club’s other centre-backs but United have set an unwanted record this season for goals conceded in the Premier League which reinforces their need for defensive options.
Varane though may be put off by a lack of Champions League football at Old Trafford.
Ole Gunnar Solskjaer is currently in a four way battle for the final two top four spots.
The club returned to the Champions League last season following a year out in which they won the Europa League.
A quarter-final exit to Barcelona though broadcasted the gulf between where United currently are and where they want to be.