Real failed last summer to lure the Belgium worldwide to the Bernabeu after the World Cup.
According to The Mail, the Blues are said to value Hazard in excess of £100m but Los Blancos are hoping to reduce the valuation to some extent by proposing a part-exchange deal.
A quote from Hazard recently hit the headlines, with the attacker telling French TV station RMC: "I know what I am going to do".
"Right now he is waiting for the club to decide upon signing him, and he has refused to renew his deal with Chelsea".
Losing an asset as valuable as Hazard - deemed one of the best attacking stars in the world - would not be countenanced at Stamford Bridge.
Manchester United and Manchester City are both linked with Fiorentina forward Federico Chiesa, who has also attracted interest from Inter, Juventus, Bayern Munich and Real Madrid.
Should Hazard leave, Chelsea would need to snap up a replacement and Richarlison could be an option with the 21-year-old having been on the club's radar since he started his career in his homeland with Fluminense.
The Belgian revealed earlier this week that he'd come to a decision over whether he would sign Chelsea's new contract offer, reportedly worth £300,000-per-week.
Hazard's current contract will expire on June 30, 2020 and he has never hidden the fact that his dream is to play at the Estadio Santiago Bernabeu.
He has given many hints that he will be moving at the end of the season.
"For now, the only thing to think about is playing well for Chelsea".