According to AS (h/t Daily Star's Rhys Turrell), sources within the Real organisation confirmed the two sides have been talking, while the Sun (h/t Daily Star's Colin Harvey) said James Rodriguez could be used in a swap deal.
Hazard was one of the Premier League's top performers last year, winning the PFA Player of the Year award and playing an instrumental role in Chelsea's run to the English title. His form in 2015-16 has been the exact opposite, as he's suffered through the worst season of his professional career.
Per WhoScored.com, the 25-year-old still hasn't scored a single Premier League goal this season, and Bleacher Report UK couldn't help but take a shot at his shooting form recently:
The Blues struggled as a team under former manager Jose Mourinho, but while most players have rediscovered their form in the wake of his departure, interim manager Guus Hiddink has failed to get the best out of Hazard.
The Belgium international continues to look disinterested at times, and he seems devoid of the confidence that drove his fantastic 2014-15 campaign.
But his dreadful form hasn't stopped the transfer rumour mill from doing its thing, and he and Real manager Zinedine Zidane haven't exactly helped matters. The two have periodically showered each other with compliments, and Hazard continued the back-and-forth in an interview with Sudinfo.be (h/t Independent's Samuel Stevens) in February:
It's always nice to hear that a player that I admired during my childhood talks about me.
I always liked Zidane. Now I know the player by heart but don't know the coach at all, even if he's started very well at the Bernabeu. I'm also still under contract at Chelsea and I feel very good there.
[On if he would want Zidane as his coach one day] Yes of course. When I was little, I watched him on television and the internet for hours.
I talked Zidane and I ate Zidane.
None of that means a transfer is imminent, but it's clear Hazard and Zidane would like to work together. It's even possible an eventual transfer to the Spanish capital is already on Hazard's mind, and that's partly what's slowing him down right now.
Het Laatste Nieuws' Kristof Terreur doubts it, however. He believes the Blues have no interest in selling their star forward, and he's not thinking about leaving west London yet, either:
Whether Hazard departs Stamford Bridge this summer or not will likely depend on two things: His performance at the UEFA Euro 2016 tournament and Chelsea's incoming new manager.
It's hard to see any club spending big money to bring Hazard in after the season he's just had, but if he redeems himself with the Belgian national team, he might be able to convince clubs his poor showings were Chelsea's fault and not his.
If he disappoints in France, perhaps the only club willing to splash the cash on Hazard would be Los Blancos, who will likely spend big during the summer, knowing full well their impending transfer ban could see them unable to register new players for the next year, per Ed Malyon of the Daily Mirror.
But the Blues' next manager will also play a huge role, and unless that man has already started wheeling and dealing behind the scenes, it's hard to believe Chelsea would effectively sell one of their most valuable players before giving the next boss the chance to work with him and have his say.
If Hazard does leave Chelsea, Rodriguez would be a sensational replacement. Like Hazard, the Colombian is a special talent who has struggled the past season, but he still has the potential to become a world-class attacking midfielder. If a swap deal is possible, it's an option the Blues have to entertain.