An expensive summer overhaul at the Bernabeu is rumoured to have been one of the conditions laid out when Zinedine Zidane returned as manager last month.
Hazard has been a long-term target for the reigning European champions, and speculation that he could leave Stamford Bridge has only increased after Chelsea failed to agree a new contract to extend his current deal beyond 2020.
According to The Sun, an opening offer of £85m (€98m) for Hazard is coming from Madrid, and it's claimed that an 'official bid' will be made this week. Marca even claims a deal could potentially be 'finalised in the coming days' to make the 28-year-old Real's third summer signing already.
Assuming the clubs can agree on a formal fee, Hazard will apparently be offered a six-year contract to make the move to the Spanish capital, where he would reunite with friend and international colleague Thibaut Courtois, who completed the same transfer last year.
Hazard's expiring contract puts Chelsea in a difficult position, with the club forced to sell this summer in order to avoid losing their most valuable asset as a free agent in a year's time.
But Real may not stop at pinching just Hazard from the Premier League, as The Sun also claims that a sub-par performance against Barcelona in the Champions League this week hasn't dampened interest in Paul Pogba, with Real even 'accelerating their pursuit'.
However, a conflicting report from Marca notes that while Pogba and agent Mino Raiola are 'happy' for a deal to go ahead, Real view the €150m (£130m) starting price as 'excessive' and simply 'don't share the enthusiasm' for Pogba that they have for Hazard.
Certainly, as far as Manchester United are concerned, the stance is that Pogba is not for sale.
"There might be a campaign, there might be press, you never know what there is. In my talks with Paul he has always been positive here and I don't see a difference in him when I chat to him," United manager Ole Gunnar Solskajer said on the matter last week.