While Gareth Bale was at pains to point out after Real’s record 12th European Cup win that he wasn't keen on a switch to Manchester United, Hazard decided it was the right time to declare he was open to offers.
The Belgium winger played a key role in the Blues’ title-winning campaign, scoring 16 goals in 36 League outings.
That has caught the eye of Real, who won the Champions League trophy with a 4-1 win over Juventus in Cardiff on Saturday night.
Asked if he would like to see an offer, Hazard said: “Of course, that’s all. We all dream. Of course, I would consider it if Madrid made an offer.
Eden Hazard admits he would find it hard to turn down a move to Real Madrid
I would consider it if Madrid made an offer
“I am keen to win trophies. It would be great to win the Champions League. But Chelsea’s goal is to win the Champions League as well.
“Honestly, I do not know what will happen. I only know that I still have a contract with Chelsea for three seasons and that we just had an amazing campaign. We want to continue on this path and want to build on this in the Champions League.
“I am feeling very well at Chelsea. But you never know what will happen.
“I have not met with the board yet to discuss a new deal. I have two games with Belgium coming up first and I am off on my holiday after that. I want to think about football as little as possible.”
Most expensive football transfers Fri, January 1, 2016
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The top 15 most expensive football transfers of all time including Gareth Bale, Raheem Sterling, Neymar and Cristiano Ronaldo
In contrast, Bale says he couldn’t be happier at Real – a bitter blow to United boss Jose Mourinho, who has already had to ditch plans to land Antoine Griezmann from Atletico Madrid.
“I have signed a long-term contract at Madrid,” said Bale. “My family is happy and I am happy so we will continue what we are doing. We are winning trophies and I am happy.
“That’s three Champions League trophies in four years and not many people can say that. It’s great to be doing that and I want it to continue. It’s why you’re a footballer and it’s the reason I came here in the first place.”
The former Southampton and Spurs star says his sole focus is on rest and recuperation to shake off all the effects of ankle surgery.
Gareth Bale has ruled out a move to Manchester United
Antoine Griezmann won't be leaving Atletico Madrid this summer
Bale came on as a late substitute in the win over Juve and admitted he thought he might be left out of the squad altogether.
He said: “It will be fine in the long term, it just needs to be given recovery time. There is scar tissue and all that and it takes time to get rid of it.
“It needs a good rest now for the summer and some good rehab on it and then it will be fine for next season.”
United pulled out of the Griezmann deal after deciding they needed a player who could guarantee more goals.
That shift has been prompted by the decision not to take up the option for an additional season in Zlatan Ibrahimovic’s contract, with the Swede sidelined until the New Year by cruciate ligament surgery.
But with Atletico banned from signing players in the summer Griezmann insisted he would have knocked back a move to Old Trafford anyway.
He said: “Atletico cannot recruit. It’s a hard time for the club. It would be a dirty move to leave now. We talked to the club’s leaders and I will be back for next season.”
With Bale also out of the frame Mourinho must now decide which striker from Real’s Alvaro Morata, Everton’s Romelu Lukaku and Monaco’s Kylian Mbappe he will turn his attention to next.