Los Blancos are said to be desperate to bring the 28-year-old Belgian to the Bernabeu as they start rebuilding under Zinedine Zidane.
However, the Blues may be forced to cash in on their talisman this summer or risk losing him for nothing when his current contract expires next year.
Madrid are hoping to force Chelsea’s hand to sell the Belgian who has publicly admitted his interest in playing for the La Liga giants.
“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,” Hazard told The Sun in a January interview when asked about a transfer to Madrid.
“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.” He added.