After returning from an ankle injury that had him sidelined for the first few games of the campaign, he's helped drive his club back to the top after a rocky start.
It would make him the highest paid player in the history of the club but, like goalkeeper Thibaut Courtois, Hazard wants to wait until the season is over.
Hazard, 26, is happy at Stamford Bridge but is reluctant to extend a contract that has two and a half years to run as it would effectively end his hopes of joining Real.
Antonio Conte believes Eden Hazard's performance against Newcastle is testament to the fact, that he is enjoying himself in his new role at the club.
Speaking after Saturday's 3-1 win over Newcastle United, Conte said: 'The club knows very well the importance of these two players, I think these two players are very happy to play for this club.
That's because they're hoping that his current contract situation will attract interest from Real Madrid.
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The forward has shown his class yet again this season, having scored a total of eight goals in 18 matches and just as importantly set up six for his team-mates.
'It is normal during the season to have the contact between the player - or the agent of the player - and the club.
Of course, if his plan works and an offer does come in, Hazard may find it very tough to reject a move to the European champions and become the next 'Galactico'.
'I will be very happy when they can have this deal.
"They are very young players and it will be great to keep them with us".