Hazard has entered the final two years of his deal and attempts at renegotiations have not resulted in an agreement.
Chelsea are keen to tie the Belgium star to a new contract after a blistering start to the season, with seven goals in eight league games.
But after selling Cristiano Ronaldo in the summer, Real Madrid have space in their attack for a new ’Galactico’ and have made no secret of their interest in Hazard.
According to Belgian publication HLN, Chelsea told Hazard and his representatives that they want talks before he left for international duty with Belgium last week.
Hazard returned to Chelsea on Wednesday and the report says the club are waiting to hear from the player.
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Chelsea are willing to make Hazard the highest paid player in their history on £300,000-a-week and it is claimed the attacker “knows his value”.
But the report adds that Hazard wants a “new challenge” and to fulfill his “childhood dream”, which is to play for Real.
Speaking at a football business event in Doha, club chairman Bruce Buck said they will do whatever it takes to convince Hazard to stay.
“Every Chelsea fan in London and around the world loves Eden Hazard,” Buck told beIN Sports.
“And he’s not only a great footballer but I think he’s a wonderful lad.
“We want him to stay and we will do what we have to do to entice him to stay.”
If an agreement is not reached by next summer, Hazard will have 12 months left on his deal, similar to the situation with Thibaut Courtois.
Courtois eventually left Chelsea and joined Real but Buck sounded more optimistic about Hazard.
“Let’s see how it goes between now and the end of the season,” he added.