Los Blancoshave long been linked with the Belgium international, who is showing no signs of extending a contract that is set to expire in 2020.
Telegraph Sporthas reported that Madrid have already opened talks with the Stamford Bridge hierarchy about Hazard and will make another approach in the next few days.
The Spanish giants are keen to get the transfer agreed early, but the Premier League side are demanding £100 million for their star player. Real Madrid don’t want to go above £85 million, according to the report.
However, the 28-year-old is understood to be hopeful that the move will be arranged, which would make him the first new signing at the Bernabeu since Zinedine Zidane’s return as coach.
Hazard has scored 14 goals in 31 Premier League appearances this season for Chelsea, who face West Ham at Stamford Bridge on Monday night.