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.
The winger, 27, has just 18 months left on his current deal at Stamford Bridge and admitted in December that contract talks had been put on hold.
Hazard was then reinstalled at the top of Real Madrid’s shopping list after cooling their interest in Neymar and Kylian Mbappe with the Belgian fuelling reports.
Chelsea have now responded by holding discussions with Hazard's representatives about his penning an extension on his current contract, according to Le10 Sport.
The Blues have maintained hope all season Hazard would stay at Stamford Bridge but so far, an offer worth more than £300,000-a-week has not tempted the forward.
In an interview with RMC Sport, the Blues star confirmed he will remain in West London until the end of the season but after that, things are unclear.
Hazard said: "As I said, I'm already finishing this year with Chelsea, I still have a year of contract after this year, we'll see.
"There was a bit of talk [with Chelsea about a new contract] but then it stopped, now we're waiting.
"Me, you know me, I've always loved Real, even before [Zinedine] Zidane, we'll see what happens."
Former Blues skipper John Terry recently suggested his old team-mate could be persuaded to stay provided Chelsea 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.”
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