Matt Hughes of the Times reported the Blues are willing to make Hazard the highest-paid player in their history, but he's hesitant to pen an extension "that would effectively end his hopes of joining Real."
The Belgium international has long been linked with a move to the Estadio Santiago Bernabeu, touted as a potential successor to Cristiano Ronaldo, who is set to turn 33 in February of next year.
Hazard, 26, has been back close to his best for Chelsea so far this season, and WhoScored.com recently pointed to one statistic that might draw Real's attention back his way:
Eden Hazard: Of the 37 shots Hazard has attempted in the Premier League and Champions League this season, he has missed the target with just 6 pic.twitter.com/WGrownIMsC— WhoScored.com (@WhoScored) December 4, 2017
MailOnline's David Kent reported Los Blancos manager Zinedine Zidane "has long admired" Hazard, whose current deal at Stamford Bridge isn't due to expire until the end of the 2019-20 campaign.
It's now said he'll make his decision on a new contract at the end of this season, similar to club and international team-mate Thibaut Courtois, who is also reported to be stalling over an extension.
Hazard is understood to be happy in his current setting, but it appears the prospect of tying himself down to new terms is premature as things stand, with the winger presumably wishing to wait and see how this season pans out before agreeing new terms.
Chelsea manager Antonio Conte recently commented on Hazard's place among the world's elite and said his charge must do more to match some of his peers, per Kristof Terreur of Belgian newspaper Het Laatste Nieuws:
Conte on Eden Hazard: “You are at the top when you win the Ballon D'Or, the Champions League, the World Cup or the European Championships. In that moment you are at the top otherwise you are a great player, but only a great player.” #cfc pic.twitter.com/KhNY6gkRbX— Kristof Terreur 📰 (@HLNinEngeland) December 4, 2017
It also helps Hazard's case that Real are "planning a squad overhaul" at the end of this season, having lost some of their attacking power this term after selling Alvaro Morata to Chelsea and sending James Rodriguez on loan to Bayern Munich.
Former Liverpool midfielder Danny Murphy recently paid Hazard major praise when he appeared on BBC 5 live Sport:
Chelsea have no immediate fears of Hazard leaving west London with another two-and-a-half years left to run on his contract, although an extension would safeguard his place at the club for seasons to come.
Real would be considered one of the few teams capable of tempting Chelsea's best away from the English capital, but a fruitful season will help convince their talisman to prolong his Stamford Bridge stay.