Hazard has been strongly linked with a move to the La Liga giants since last summer.
The Belgian has previously admitted that he wants to play for Real Madrid one day though has remained tight-lipped in recent weeks.
Real are still looking for a replacement for Cristiano Ronaldo, who departed last summer as he joined Juventus.
Hazard is believed to be Zinedine Zidane’s No 1 target for the summer though the transfer fee could cause issues.
But the Independent now claim that both clubs want to resolve Hazard’s future.
It’s reported that negotiations have not been smooth but that a deal is edging closer.
Indeed, a signing is that near that Hazard is reportedly on the verge of agreeing personal terms with the La Liga outfit.
Hazard appeared to be saying his goodbyes to the Chelsea faithful at Stamford Bridge on Thursday.
Chelsea progressed to the Europa League final by beating Eintracht Frankfurt on penalties.
Hazard celebrated wildly with his team-mates after the final whistle but took some time to reflect on his own and looked emotional as he clapped the adoring crowd.
The Belgium international also threw his shirt into the crowd in what could be his final home game.
But Hazard is adamant his future is not on his mind.
“I don’t think [about] that,” he said when asked if that was his final home game.
“I’m only thinking about winning something for this squad, these fans.
“If it [the final] is my last game I will try to do everything, if it’s not, then I will see.
“In my mind, I don’t know yet, so that’s it.”