The Blues winger has been one of Antonio Conte’s top performers this season.
Hazard’s 11 goals and four assists in the Premier League have helped push Chelsea to within touching distance of the title.
And with the 26-year-old happy at Stamford Bridge, Chelsea want to tie him down long-term.
Eden Hazard is discussing the possibility of signing a new contract at Chelsea
Hazard is in negotiations with the Blues over a new deal
Zinedine Zidane is said to have earmarked the Belgium international as a key target at the Bernabeu this summer.
And Madrid are apparently ready to offer James Rodriguez and Alvaro Morata as part-exchange for Hazard and Chelsea team-mate Thibaut Courtois.
However, according to The Guardian, Hazard is in negotiations with the Blues over a new deal.
Antonio Conte will hope Eden Hazard stays at Chelsea and out of the clutches of Real Madrid
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Zinedine Zidane wants to sign Eden Hazard this summer at Real Madrid
The Chelsea star has three seasons left on his current contract in west London.
But Roman Abramovich wants to make it difficult for Madrid to lure him away at the end of the campaign.
Madrid are claimed to be concerned about Eden Hazard’s Chelsea talks and are keen for him not to extend his deal.
But the La Liga giants do believe he would welcome a summer move.