The Belgian has always been a long-term target for the Spanish giants who are looking to spend big this summer after a poor domestic campaign.
Hazard, 28, will have just one year left on his Chelsea contract this summer and the London Evening Standard claim Los Blancos are ready for a summer tilt at the Blues star.
Paris Saint-Germain duo Neymar and Mbappe have long been regarded as Real’s first-choice targets as they look to challenge bitter rivals Barcelona next season.
However, the French giants are not prepared to sell any of their star players as they look to become kings of Europe for the first time in their history.
Add the pair’s wages, Mbappe is on £160,000-a-week and the Brazilian incredibly earns more than £500,000-a-week after tax, it looks like the numbers do not add up for Barca.
Hazard is the cheaper and easier option, with the Belgian already going on the record about his desire to one day make the Bernabeu Stadium his home.
Chelsea have maintained hope all season Hazard would see his long-term future at Stamford Bridge.
But even the prospect of a new lucrative new deal, worth more than £300,000-a-week, has not tempted the forward to put pen to paper.
And his interview with French radio station RMC earlier this month was taken as a massive hint he was off to the Spanish capital next season.
He said: “I know what I am going to do. I have made a decision.”
But former Blues skipper John Terry provided a glimmer of hope for Chelsea fans, suggesting his old team-mate could be persuaded to stay in west London – provided they reach the Champions League next season.
Terry told Sky Sports: “Any player wants to see security in the manager and the players you bring in.
“Is the Champions League a big thing for him and the football club? Yes, it probably is.
“Can we do it (qualify for the Champions League)? Yes, we can.
“He’s up there with (Lionel) Messi and (Cristiano) Ronaldo.
“He loves London, he loves the football club and wants to stay. He also wants to win trophies.
“Can he do that at Chelsea? Yes, he can. His family love London. He wants to be a Chelsea player, that’s for sure.”
Chelsea will look to stay in the hunt for a top-four finish when they face Tottenham at Stamford Bridge tonight.
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