According to Le 10 Sport (h/t Metro), the Spanish giants have struck the "first agreement" with the Premier League side over a possible transfer of the Belgium international, potentially pushing Hazard closer to a switch to the Santiago Bernabeu outfit.
As relayed by Metro, it has also been reported by Gianluca Di Marzio of Sky Sport that Hazard's lawyers, the player's father and Real Madrid representatives will meet on Saturday to discuss a possible move.
According to Le 10 Sport, the player himself has already held preliminary discussions with Real about a possible switch. As noted by Metro, Hazard's Blues team-mate Thibaut Courtois has also been strongly linked with a move to the Spanish capital this summer.
Kristof Terreur of Het Laatste Nieuws reported a fee hadn't been agreed, although discussions were on the horizon:
Kristof Terreur 📰 @HLNinEngeland
A colossal £170million transfer fee hasn’t been agreed with Chelsea as yet, despite one report in France (that will tell you something else in 6h). Real Madrid still exploring the Eden Hazard transfer with meetings planned, but what will Chelsea do? #cfc https://t.co/9foLDUO8MN
Hazard has two years left to run on his contract at Chelsea and has been tipped as a player with the potential to fill the quality void left by Cristiano Ronaldo's departure at the Bernabeu. The Portuguese moved to Juventus earlier in the window.
At the end of Belgium's FIFA World Cup campaign, which culminated in the team finishing third, Hazard was coy about where he'd be playing his football in 2018-19, courtesy of ITV Football:
Chelsea supporters will be desperate for it to be in London, as the club are heading into a vital campaign under the guidance of new manager Maurizio Sarri.
While the Blues won the FA Cup last season under Antonio Conte, they were well short of their usual standards in the Premier League and subsequently failed to qualify for next season's UEFA Champions League.
That fact will only add fuel to the rumours surrounding Hazard, as a player of his calibre should be playing at the highest level. Plus, if he is keen on a move to Madrid, it's unlikely he'll be extending his contract with the Blues any time soon.
It would be huge blow to Sarri in his first season in charge if Hazard was to leave, as the playmaker offered up a reminder of how effective he can be at the World Cup, bringing a vibrancy and incision to Roberto Martinez's Belgium outfit.
These figures sum up just how important he was to the Red Devils' cause:
If Madrid do go hard after Hazard it leaves Chelsea with a big decision to make, as they will not want to be in a situation where the player only has one year remaining on his deal in 12 months' time. In a summer of so much change for the Blues, the decision on their No. 10 may be the most significant of all.