Los Blancos are said to be "willing to pay the second-biggest fee in transfer history" for Hazard. Chelsea refuse to sell and believe he can be convinced to sign a new deal, according to John Richardson and Tom Hopkinson at the Sunday Mirror.
Hazard has two years left on his current deal, and Chelsea plan to offer him a new contract worth £300,000 a week. They will begin negotiations when he returns to the club from holiday.
Maurizio Sarri has said he hopes to improve Hazard during his time with the Blues, as shown by Goal:
Hazard cast doubt over his Chelsea future after the 2018 FIFA World Cup where he helped Belgium secure third place in Russia, per Dominic Fifield at the Observer:
"After six wonderful years at Chelsea it might be time to discover something different. I can decide if I want to stay or go, but Chelsea will make the final decision if they want to let me go.
"For my career, certainly after this World Cup, I’m at a moment where I might change. We all know that I’ve always felt well at Chelsea, so … I’ve been linked to all the big clubs. Next week it will be Bayern. No, I’m joking. You all know my preferred destination."
The 27-year-old enjoyed a strong World Cup where his performances with Belgium caught the eye. He's widely considered to be one of the most skilful players in English football and has won two Premier League titles, the FA Cup and the UEFA Europa Cup with the Blues.
Hazard has previously said playing for Real Madrid is "everyone's dream," per beIN Sports (h/t ESPN FC's Dermot Corrigan).
Real Madrid look in need of some added attacking flair after losing talisman Cristiano Ronaldo to Juventus. However, there are no meetings planned with Chelsea yet, according to Simon Johnson at the Evening Standard:
The Spanish giants would be able to offer Hazard UEFA Champions League football and also look to be one of the few clubs who could tempt him away from Stamford Bridge.
Sports journalist Robbie Dunne said Sarri's appointment as Chelsea boss is bad news for Real Madrid:
Sarri is known for producing exciting, attacking teams and Hazard should thrive under the Italian. The Blues are rarely bullied in the transfer market but will want their star man to sign a new contract and commit his future to the club.