The Belgium international has been linked with a move to Spain for some time, but has seen speculation regarding a big-money switch stepped up over recent weeks.
Hazard has continued to leave the door open to a new challenge, with the 27-year-old having spent the last six seasons in England.
Former Chelsea midfielder Burley feels the time is right to make a fresh start, with Portuguese superstar Ronaldo having shown when leaving Manchester United for Madrid that La Liga can be a perfect springboard to becoming the world’s best.
Burley told ESPN FC: "At the end of the day, he has been there six years.
"It has been very fruitful for him, changes are afoot at Chelsea, Sarri is a very good coach and plays attacking football.
"But can any player really turn down Real Madrid if they come knocking at the door?
"And you are a top player and want to go enhance your game like Ronaldo did.
"When Ronaldo left United, it propelled him and I think it is time for Hazard to propel himself to that next level."
Hazard has made 300 appearances for Chelsea, scoring 89 goals.
His latest comments on his future have hinted that he will soon be heading elsewhere, with the Madrid rumours fuelled after helping Belgium to a third-placed finish at World Cup 2018.
Hazard told reporters: "After six wonderful years at Chelsea it might be time to discover something different.
"Certainly, after this World Cup, I can decide if I want to stay or go, but Chelsea will have the final decision.
"You know my preferred destination."