Chelsea are allegedly demanding £150 million for the Belgian, but the Spanish side could offer a part-exchange deal including Mateo Kovacic.
According to Onda Madrid, as cited by AS, the two clubs have reached an agreement for the 27-year-old.
But the La Liga giants are not willing to meet Chelsea's valuation of Hazard, who will be out of contract in 2020.
Real Madrid are allegedly willing to make Kovacic's loan deal to Chelsea permanent in order to sign the Belgian.
The Croatian midfielder is currently on a season-long loan at Stamford Bridge and he's made an impressive start to life in the Premier League.
Hazard was linked to Real Madrid this summer and he has admitted that it's his "dream" to play for the club.
But he has insisted that he will never force a move to Spain after six successful years with the Blues.
Maurizio Sarri is not keen to sell his star player and the Blues are hoping to pin him down with a new £300,000-a-week deal.
But he is reportedly waiting to see if his switch to Solari's side materialises before he puts pen to paper.
Sarri has told Hazard he only wants him to sign a new deal with Chelsea if he is committed to the cause.
He said: "You know very well that Eden is very important for us, like every special player.
"It's up to him. I think the club is available to sign a new contract immediately, but it's up to him if he wants to renew or not. I don't know."
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