As reported by Pete Jenson and Ian Herbert of the Daily Mail, Hazard has long been linked with a move to the Santiago Bernabeu, and while he has not shut the door on the possibility of staying at Stamford Bridge, he said he could be tempted by the switch.
"Real Madrid could be interesting for me as everyone knows," he told L'Equipe (h/t Marca). "They know what they have to do if they want me."
He added: "If it is going to Madrid just to go there then no. I need a project and at Chelsea I have one."
Hazard wants to see the uncertainty over Blues boss Antonio Conte's future resolved before he makes a decision on his own: "To say that I am going would be too easy and so first of all I want to see what will happen at Chelsea ahead of next season. I want to see if the coach stays or not."
Chelsea put up a disappointing title defence in the Premier League last season and only managed to finish fifth, meaning they will not be in the UEFA Champions League next year.
Along with Conte's future at the club being up in the air, Hazard also wants reassurances that the Blues will make efforts to ensure a more successful campaign next season.
"If I stay then it would be because they are going to improve on last season," he said. "I am not going to remain if they are going to be worse."
The Belgian was Chelsea's strongest performer, and he finished the campaign with 17 goals and 13 assists in all competitions.
Hazard's numbers don't always compare that well to those at the top of the game, such as Real's Cristiano Ronaldo—he has scored 20 goals in a season just once in his career and that was at Lille—but he contributes a lot even when he fails to find the net, per Squawka Football:
The winger's biggest strength is his dribbling, and he's electrifying with the ball at his feet:
Real's front line could use freshening up now that Ronaldo is 33 and Benzema, 30, scored just 12 goals in all competitions last season.
Hazard would fit in well there, and he's open to going in the right circumstances, but much will depend on Chelsea's activity in the coming weeks.