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According to French media outlet RMC Sport, an agreement has been reached between Hazard and Real, but Chelsea are yet to accept a bid from the European champions.
After several attempts to get Real Madrid to approach the London side to broker a deal done, Hazard dropped the biggest hint yet that he might not stay at Stamford Bridge past the summer.
Speaking after the third-place playoff against England, he told Belgian outlet HLN: "After six wonderful years at Chelsea it might be time to discover something different."
"Certainly after this World Cup. I can decide if I want to stay or go, but Chelsea will make the final decision - if they want to let me go. You know my preferred destination."
Hazard has also previously said he would wait for clarity over the future of former boss Antonio Conte before making a decision over his future. Chelsea appointed Maurizio Sarri as their new head coach on Saturday.
He added: "We knew already for a while that Sarri was going to become the new manager. If he's a good manager? If he lets me win trophies, yes."
"Dries Mertens has told me he's a good coach. We'll have to work hard. The Italian way, but we were used to that with Antonio too."
The 27-year-old has two years left on his contract, and the club will be forced to sell him if he continues to stall over a new deal.
Madrid are reportedly ready to spend £200m to bring Hazard to the Bernabeu in order to fill the void left by the departure of Cristiano Ronaldo to Juventus.
RMC Sport go on to report that Hazard is keen on taking over Cristiano Ronaldo's throne at Real. The Belgium international, who has a verbal agreement in place, has always stated that he wants to win the Champions League - a competition domination by Los Blancos in the past 5 years.
Will Eden Hazard stay at Chelsa past the summer? Let us know in the comments section below.
