The Belgian playmaker has signed a six-year contract with the club and will be unveiled in Real Madrid white on Thursday, subject to a medical.
His latest trail of class was quiet at display in the Europa League final against Arsenal.
Eden Hazard, 28, boasted of one of Europe's best goal involvement stats in 2018/19 with 21 goals and 17 assists in 52 games.
The Belgium global has been at the centre of transfer speculation for most of the last year as he prepared to enter the final year of his contract at Stamford Bridge.
"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", she said.
Hazard is obviously looking forward to a new chapter in his life and was respectful also to the Chelsea fans and his club, he had this to say: "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".
Terreur told Sky Sports' Transfer Talk podcast: "I expect him to play on the left because I know there has been a meeting in 2016 between Real Madrid and Hazard's people where they explained to him that they saw him as a long-term replacement for Cristiano Ronaldo playing on the left wing, where he's playing for Chelsea basically".
Hazard joined La Liga giants Madrid from Chelsea on a five-year deal on Friday for a fee of €100 million (£88m/$112m), but with incentives could rise to as much as €140m (£124m/$146m).
And Terreur expects the 13-time European Cup winners to pull out all the stops for Hazard's official unveiling next week.