The 27-year-old has less than 24 months to run on his current deal following negotiations with the west London side over a new long-term deal.
While Hazard’s current deal has significant time left to run on it, the Belgium international could leave for nothing at the end of the 2019-20 season unless he commits to a new contact.
The Chelsea FC number 10 saw his Belgian team-mate Thibaut Courtois secure a move to Real Madrid in the summer after the Blues decided to cash in on the goalkeeper rather than lose him for nothing.
Hazard has been regularly linked with a move to the Champions League holders over the past few seasons but a deal has failed to materialise despite ex-manager Zinedine Zidane being an admirer.
However, Real Madrid do have a void to fill up front after Cristiano Ronaldo moved to Juventus from Real Madrid in a £100m deal in the 2018 summer transfer window.
Ronaldo’s exit has coincided with Hazard’s best start to a domestic campaign following a return of seven goals in eight games in the Premier League.
Asked about his contract situation at Chelsea FC after linking up with his compatriots for the international break, Hazard is quoted as saying by BBC Sport:
“I don’t want to say, ‘yes, I am signing a new contract’ and then in the end I don’t end up signing.
“I want what’s good for me, but I want what’s good for the club because the club has given me everything.
“So I will see. Sometimes in my head, I wake up in the morning and think I want to go. Sometimes I think I want to stay.
“It is a hard decision. It is my future.”
Hazard broke the deadlock in a 3-0 win over Southampton before the Belgium international created Ross Barkley’s first goal for Chelsea FC.
The Chelsea FC number 10 has won two Premier League titles, the Europa League, the League Cup and the FA Cup since his move to the Blues from Ligue 1 side Lille in 2012.
The Blues forward will face his former manager Jose Mourinho in their next Premier League fixture when Manchester United make the trip to Stamford Bridge.
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