Goal previously reported the deal is worth a minimum €100m, but can rise to €140m (£124m/$146m) with incentives.
The club announced Hazard will be presented as a Madrid player on June 13.
The Blancos sent representatives to London earlier this week to meet with Chelsea director Marina Granovskaia, and Hazard has now officially made the move to Madrid after seven seasons at Chelsea.
The 28-year-old has made no secret of his desire to join Real Madrid, calling the move a 'dream' near the beginning of the 2018-19 season.
With his contract at Stamford Bridge expiring at the end of the 2019-20 season and no progress made on an extension, Chelsea have accepted the need to cash in on their Belgian superstar this summer rather than let him walk for free.
Granovskaia thanked Hazard in a statement, saying: "Although it is with sadness we say goodbye to Eden and we made it absolutely clear to him the club wished him to stay, we respect the decision he has made to take on a new challenge in a different country and follow his childhood dream of playing for Real Madrid.
"The memories he leaves us with will not fade. He provided all who watched Chelsea play with great entertainment and many match-winning contributions, and for that we thank Eden enormously.
"He has been a model professional throughout his time at the club, a wonderful individual to have around and a joy to work with. We look forward to welcoming him back to Stamford Bridge in the future."
Hazard's departure marks the end of an era for Chelsea, with the attacker having marked himself as one of the club's all-time greats over seven seasons.
Hazard led Chelsea to two Premier League titles, two Europa League titles and one FA Cup during his time at Stamford Bridge.
He was also named the PFA Players' Player of the Year in 2014–15, and was runner-up for that award on two other occasions.
After a brilliant career at Chelsea, Hazard has been keen to move on to another challenge and the possibility of working under Madrid manager Zinedine Zidane was a major selling point for the Belgian.
Madrid are looking to rebuild after a disappointing 2018-19 campaign that saw them use three different managers and end 19 points behind champions Barcelona in the La Liga table.
The club's spending began before the season even ended with the capture of Eder Militao from Porto, while on Tuesday they announced the €70 million (£62m/$79m) signing of Luka Jovic from Eintracht Frankfurt.