Courtois officially completed his £35.5m switch to the European champions earlier today, bringing an end to a long-running saga which turned sour when the Belgian shot-stopper refused to turn up to pre-season training this week.
Courtois's international and former club teammate Hazard has also been repeatedly linked with a move to the Bernabeu this summer, and the 26-year-old admitted that he would like to see Hazard make the same switch from Stamford Bridge.
"I want to thank [the Chelsea fans] for the last four years I have been there. I had to take the chance (to come back) because my two kids are here. I want to live close to them and see them grow up," Courtois told reporters.
"I know the supporters are maybe not too happy at the moment, but I wish them all the best and I hope they remember me well.
"Eden congratulated me. He knows it is a dream for me to be here. I would like to be always by his side. He is a great player and if he could come here, it would be fantastic."
Chelsea are unwilling to sell Hazard this summer, although Madrid have until the end of the month to make a move despite the English transfer window closing at 5pm today.