With Real's Club World Cup campaign now over, Modric is leaving the Spanish giants after 11 years to join AC Milan.
The Croatia international spoke for the final time as a Real Madrid player to the club's official TV channel.
On Florentino, he said: "The president has been very important to me, first because he brought me here. Since that moment, he has always shown me special affection, he has always treated me very well.
"Now I can say it: he treated me differently, he had a very special affection for me. And I think he showed it in my last match, because I had never seen the president cry."
Real Madrid gave me everything
On his career with Real Madrid, Modric said: "It's hard to say, but they are mixed emotions. An unforgettable, glorious, and victorious era has ended. What I've experienced gives me even more joy.
"Remembering everything I've achieved here makes me very happy, even if it's over. At Real Madrid, I've grown as a player and as a person. Real Madrid has given me everything in football, and for that I'll be grateful for life. I'll always be a Real Madrid fan."
Asked for one memory from his time with Real Madrid, the veteran added: "It's difficult to choose one moment among so many precious moments I've experienced here, but I always highlight and want to remember La Décima, because that's where we can say it all began.
"It started the dominance of the last 12 or 13 years, which has been incredible, winning six Champions League titles in 10 years. I always mention La Décima because it was impressive, and the way we won it perfectly defines what Real Madrid is all about."