Chelsea are desperate to keep Hazard but the player appears to be unsettled at Stamford Bridge.
He was left out of the Blues’ 5-0 win over Stoke on Saturday amid increasing speculation over his future.
His father, Thierry, recently confided to a Belgian newspaper that Hazard is keen on a move to the Bernabeu.
And The Sun claim Real are plotting a massive £120million deal that Chelsea may not be able to refuse.
The dilemma for Chelsea is that Hazard’s contract expires in 2020.
That gives the club until midway through next season to commit Hazard to a new deal.
If he doesn’t then there is a risk of him eventually leaving for free.
And that means in the mean time his value will drop for each transfer window that passes.
Chelsea don’t want to lose Hazard for free or even on a cut-price fee.
So cashing in on the player now would seem like the obvious choice, unless they are confident he will extend his contract.
Selling the Belgium international would be a major boost for Antonio Conte’s transfer coffers and the boss is open to spending money this money.
However, the likelihood of Chelsea being able to sign a player of Hazard’s stature midway through the season could prove difficult.
The Sun claim Manchester United are also linked with a £90m move for Hazard but that move appears very unlikely.
Chelsea sold Nemanja Matic to the Red Devils in the summer and Conte is not likely to be keen on losing his best player to a Premier League rival.