Spain's AS sports daily, citing unnamed sources, said on its website that the Premier League side had accepted Real's offer for the 28-year-old.
It said Hazard, who is training with Belgium ahead of a weekend Euro 2020 qualifier against Kazakhstan, had been informed and the deal could be announced later.
Sky Sports television said Chelsea had wanted in the region of 130 million pounds, a fee that could ultimately be reached through performance-related clauses.
Hazard, who was set to become a free agent in June 2020, guided Chelsea to the Europa League title in Baku last month with two goals in a 4-1 win over London rivals Arsenal.
"My dream was to play in the Premier League and I did it for seven years. I think it's a goodbye," he said after that game.
He joined Chelsea from Lille in 2012 for 32 million pounds and won the Europa League in his first season.
He later won the Premier League title in 2015 and 2017 as well as one League Cup and FA Cup.
He scored 16 goals and created another 15 in the Premier League last season as Chelsea finished third in the table.
(Reporting by Alan Baldwin/Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
