The playmaker agreed a five-year contract with the Spanish champion, admitting he was determined to leave White Hart Lane as soon as he became aware of the interest.
"I chose the Spanish league because it's one of the best leagues in the world," he said.
"And I chose Madrid because they are the best club in the world. They've won so many things, so many titles. It's amazing for me to be here.
"When I heard that Madrid noticed me, the rest lost interest.
"There are no hard feelings towards Tottenham, whatever may be said. I'm simply grateful to them.
"This is the best club in the world, with the best coach, the best players.
"I'm ready for anything and I've come here to win a place in the starting team.
"I know that will be hard but I believe I have the qualities needed to play here. I want to help the team win titles."
Real has endured a disappointing start to the season, securing only one point from two league games and losing the first leg of the Spanish Super Cup against Barcelona 3-2.
"I've seen all three games. It's not important how you start, but how it ends," Modric said.
"There are still a lot of games to come in all competitions. The important thing is to finish well."
The money raised from Modric's departure will enable Tottenham manager
Tottenham is reported to be interested in Willian of Shakhtar Donetsk and Internacional's
Gerard Pique has declared that Barcelona must try and talk Manchester United target Thiago Alcantara into staying at the Catalan club after the midfielder netted a hat-trick in Spain's 4-2 win over Italy in the European Under-21 Championship final.