Ronaldo left Real Madrid for Juventus in a £99m move last week, leaving president Florentino Perez with a big task this summer.
Now Spanish outlet Don Balon say Hazard is the man Real want ahead of Paris St-Germain duo Kylian Mbappe and Neymar, according to the reports.
And the Belgium international has made no secret of his yearning to make a switch to the Bernabeu if Real make a move for him.
After winning the third-place game with Belgium against England at the World Cup, the 27-year-old opened up on his future.
“I will go on holiday and ask myself the right questions,” Hazard explained.
“After six wonderful years at Chelsea, it might be time to discover something different.
“I can decide if I want to stay or go, but Chelsea will make the final decision – if they want to let me go.
“For my career, certainly after this World Cup, I’m at a moment where I might change.”
The reports allege that the next few days could be key in securing a deal for the winger.
New Chelsea manager Maurizio Sarri will need to decide if his star man stays at Stamford Bridge or not.
Chelsea will begin their Premier League campaign against Huddersfield on August 11.