The Belgian has been linked with a move to Los Blancos this summer and is known to be admired by Zinedine Zidane who has returned to the Bernabeu as manager.
Hazard has stalled on contract negotiations at Stamford Bridge with many expecting him to leave at the end of the campaign.
The Daily Record claim Chelsea have rejected an initial offer for their talisman.
But Hazard has taken it as a good sign as he believes Real will go all out to sign him this summer.
The winger has been in good form for Chelsea this season, netting 16 goals and providing 11 assists.
The Blues are aware they may have to cash-in on Hazard as he is out of contract in the summer of 2020.
And if he doesn’t sign a new deal he could negotiate to leave on a free abroad in January.
Chelsea are said to believe they can still secure a fee of around €100million for Hazard, despite his contract running down.
The reports also claim Real’s pursuit of the 28-year-old is in addition to them trying to sign Neymar this summer in a blockbuster deal.
Zidane has given his approval of both deals after returning the to the club.
Hazard spoke about joining Real Madrid in January, saying: “Why not? You know it very well. You’re asking the question, but you know what the answer will be so you don’t have to ask it.
“I’ve won everything in England, apart from the Community Shield, but that doesn’t mean I’m leaving.
“I’ve always said that I wanted to explore something different after England but there are also things that might make me stay.
“After the World Cup, I wanted to leave, but in the end I stayed at Chelsea and I’m playing one of the best seasons so far.”
Chelsea return to action after the international break against Cardiff on Sunday, March 31.