Hazard has one year left on his Chelsea contract, but the Spanish side will still pay Chelsea £89m for his signature.
The Guardian report Real Madrid will announce the signing of Hazard within the next few days after agreeing a fee with Chelsea.
Real Madrid have long been linked with the Belgian forward, and Hazard appeared to say goodbye to Chelsea fans after the Europa League final.
He said: “I think it is a goodbye, but in football you never know.”
“My dream was to play in the Premier League and I did this for seven years in one of the biggest clubs in the world.
“So now maybe it’s time for a new challenge.”
Hazard will leave Chelsea after seven years, where he won the Premier League twice.
He also won an FA Cup, and two Europa League titles.
Hazard will form part of Real Madrid’s rebuilt squad, as they aim to fight back in the Champions League and La Liga next season.
Los Blancos have already signed highly-rated striker Luka Jovic from Eintracht Frankfurt for £58m, as well as Eder Militao from Porto for £44m.
The club finished third in the Spanish league last season, 15 points behind champions Barcelona, and eight points behind cross-city rivals Atletico Madrid.
They were also knocked out of the Champions League in the round of 16 to subsequent semi-finalists Ajax.
Chelsea boss Maurizio Sarri respected Hazard’s decision to leave the club, he said.
Speaking after the Europa League final last week, Sarri said: “As you know, Hazard is a wonderful player, but you have to spend two or three months to understand him as a man.
“But when you are able to understand him as a man, he’s a wonderful man.
“I knew Eden wanted to leave. But I wanted to respect his decision.”