Hazard has agreed a five-year contract and will be presented as a Real player on 13 June, subject to a medical.
The Belgian scored 110 goals in 352 games for Chelsea after joining from Lille in 2012, and scored twice as they won the Europa League final last week.
“Leaving Chelsea is the biggest and toughest decision in my career to date,” said Hazard.
Writing on Facebook, he added: “I hope you understand I had to pursue my next chapter, just as each and every one of you should when you have the chance to pursue your dreams.
“Now it’s in the open I would like to put on record one thing was always clear to me, I have loved every moment at Chelsea and not once did I ever consider, nor would I have left for any other club.”
Chelsea director Marina Granovskaia said: “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.”
In seven seasons at Chelsea, Hazard, whose Blues contract was due to expire in a year, won the Premier League and Europa League twice, the FA Cup and League Cup.
He was also named Chelsea’s Player of the Year on four occasions.
“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,” Granovskaia added.
Chelsea have been placed under a transfer ban by Fifa, meaning they will not be able to sign a replacement for Hazard.
The Blues are banned from signing players during the next two transfer windows until the end of January 2020. They failed with their first appeal to football’s world governing body, and have taken their second to the Court of Arbitration for Sport.
Hazard’s Stamford Bridge journey
When the then 21-year-old Hazard made his highly anticipated move to England in June 2012, the Blues had just been crowned champions of Europe for the first time under manager Roberto di Matteo.
The Belgian, who had also been wanted by Manchester United, hit the ground running with two assists on his league debut and went on to register 13 goals and 23 assists in his first season at Stamford Bridge.
A promising debut season was capped by a first trophy, though an injured Hazard had to watch from the sidelines as interim boss Rafael Benitez – Di Matteo’s mid-season replacement – guided Chelsea to victory in the Europa League.
Hazard won back-to-back player of the season awards following Jose Mourinho’s return to the club in June 2013 – an award he went on to win four times in six campaigns.
An improved 17 goals and 10 assists in 2013-14 was eclipsed by the Belgian’s joint-most prolific season for Chelsea the following year, scoring 19 times in all competitions and being crowned Premier League player of the season as Chelsea won a league and League Cup double.
Hazard featured in all 38 games that season, but 2015-16 proved difficult as Mourinho’s second term at Chelsea fell apart. He scored just six goals, enduring a 28-game goalless run for the Blues stretching from August until the end of January.
Antonio Conte’s subsequent arrival brought a second Premier League title, with Hazard returning to form with a record 16 league goals – a figure he matched in his final season at the club under Maurizio Sarri.
A consistent match-winner for the Blues, Hazard scored the only goal to seal an FA Cup final victory against Manchester United last season and starred in their second Europa League triumph in May.
“When I think back over my special moments in a Chelsea shirt, there are so many, we are lucky to win many more games than we have lost,” said Hazard.
“Those who know me understand it is not me to list what I have won, all I can say is each and every award both as a team and as an individual have been incredible.
“One day I will be able to sit down over a beer and talk about or laugh with pride about my missed penalty that set up my header to win the league against Palace, or those goals against Tottenham, Arsenal or Liverpool.”
Hazard makes ‘dream’ move
Hazard had made no secret of his interest in a move to Real during his latter years with Chelsea, describing a move to the Bernabeu as a “dream”.
In June 2017, the Belgian said he would “listen” if Real Madrid offered him a deal, before admitting “it might be time to discover something different” after six “wonderful years” at Stamford Bridge following the 2018 World Cup.
Later that year, Hazard admitted he was torn between signing a new deal at Stamford Bridge and heading to Madrid, and believed Real to be “the best club in the world”.
“It is my dream since I was a kid. I was dreaming about this club,” he said when asked about a potential move.
Meanwhile, returning Real boss Zinedine Zidane refused to remain quiet too, telling the media at a press conference in April that he had “always appreciated” Hazard as a “fantastic” player.
The rumours rumbled on and Sarri seemed resigned to losing Hazard following the club’s final Premier League game of the season, insisting the club must “respect” the player’s final decision.
Hazard revealed then that he had informed the club of his decision “a couple of weeks ago”, eager for his future to be resolved as early as possible.
He is Real’s third signing after the arrivals of striker Luka Jovic from Eintracht Frankfurt for a fee reported to be in excess of £50m and defender Eder Militao from Porto for 50m euros (£42.7m).
Real finished third last season in the league, 19 points behind champions Barcelona, and having won the Champions League for three successive seasons, they were knocked out by Ajax in the last 16.
What are Chelsea losing?
Chelsea’s transfer ban leaves the club unable to sign players until January 2020. And even if they could, they would face an enormous task filling the void that the Belgian will leave.
In addition to his 110 goals, he also made 91 assists for Chelsea. Eighty-five of those goals and 54 of the assists came in the Premier League – only Frank Lampard (147 goals, 90 assists) and Didier Drogba (104 goals, 55 assists) have been involved in more Blues goals in the English top flight.
Hazard was involved in almost half of all Chelsea’s league goals during the 2018-19 season – 49.2% – scoring 16 times and making 15 assists as the Blues achieved their goal of a top-four finish.
His 16 goals accounted for 25.4% of Chelsea’s total in the competition, while he also finished top of the league’s assist chart.
“I hope that we are drawn against each other in the Champions League next season and every season so we can meet again.”
‘One of Chelsea’s all-time top Premier League players’ – analysis
However, as the season went on, it became clear the Belgian was determined to pursue his dream move to the Spanish capital.
Once it became obvious Hazard wanted out, senior executives at Chelsea determined they would work with the player to facilitate his move.
Unlike the acrimony that accompanied Diego Costa’s departure 12 months ago, Hazard has never caused the club any issues.
His two-goal contribution in the Europa League final victory against Arsenal – in what turned out to be his final Chelsea game – cemented his status as one of the club’s all-time top Premier League players.