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."
Real Madrid are paying an initial 100m euros (£89m) for the attacking midfielder, though bonus payments could see Chelsea receive at least another £60m from the sale of the player.
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.
BBC Sport
More about: EdenHazard
