The Belgian has been strongly linked with the La Liga giants having repeatedly hinted at a desire to move to Spain since the World Cup.
Hazard will enter the final year of his Stamford Bridge contract at the end of the season and has yet to show any sign of committing to a new £300,000-a-year deal.
Asked about his future last month, he responded: “Sometimes in my head, I wake up in the morning and think I want to go. Sometimes I think I want to stay. It is a hard decision.
“Real Madrid is the best club in the world. I don’t want to lie today. It is my dream since I was a kid. I was dreaming about this club. We will see.”
Hazard is currently waiting for a signal from Los Blancos that they will firm up their interest with an offer next summer.
The European champions are shrouded in uncertainty following the recent sacking of Julen Lopetegui, having yet to name a permanent replacement with Santiago Solari in temporary charge.
But Real’s lack of a talisman since Cristiano Ronaldo’s summer departure to Juventus means it is likely they will chase a big-money addition in 2019.
And former Chelsea midfielder Michael Ballack insists the Londoners must talk to the 27-year-old in order to convince him he is best served by staying with Maurizio Sarri’s charges.
“You have to pay him a lot of money!” Ballack told Goal when asked about what Chelsea must do to keep Hazard.
“But more importantly, you have to give him a vision. You have to tell him how important he is, ask him what he wants from the club and to give him a little bit more responsibility.
“You need to find out what he wants as well, ask him if he’s happy. Of course, you want to win something and be as successful as possible as an individual player.
“It is normal to look around and think, ‘With what club can I achieve that with?’
“Everyone who loves Chelsea wants Hazard to stay.
“Chelsea is an ambitious club and it will always be possible that a coach and the players are competitive at the highest level.
“Only he can give the answer of what he wants for himself, his ambitions for the club and give signs.”
“I always said he is an incredible player. From my perspective, he should have even more responsibility and even more impact on the team, not just to enjoy individual freedom which he likes.
“Players sometimes don’t want to have this heavy weight on their shoulders to carry the team or to get asked in certain situations to have that responsibility.
“I think he is the player who should do that in that team. If you understand how important it is for yourself and for your game and you can handle it, then it will transfer to the team and you will take the next step.”