European champions Real are ready to offer cash plus £40m-rated striker Alvaro Morata to land the 25-year-old Belgium superstar.
The Spanish giants are also aware that Chelsea are keen on another of their players, Colombia star James Rodriguez.
Their manager Zinedine Zidane is a massive admirer of Hazard, and the winger is as big a fan of the legendary French midfielder.
Chelsea want to keep Hazard despite a poor season, in which the form that had made him Footballer of the Year 11 months ago almost totally disappeared.
The Stamford Bridge top brass believe he could yet recapture those heights under new boss Antonio Conte, who will join after Italy’s involvement in Euro 2016 ends.
But a huge offer, in addition to the chance to sign one of their major summer targets in Morata, could convince them to sell.
Morata, who has been on international duty at the Euros, has warned, that he does not want to return to the Bernabeu to sit on the bench.