After intense negotiations, Spurs revealed that they had agreed to sell the Croat playmaker to the Santiago Bernabeu side for a fee believed to be £33 million.
The diminutive midfielder has thanked the north London club but is now looking forward to a new chapter in his career.
“I have to work very hard to earn a spot in the starting eleven and I’m prepared to do it,” Modric told reporters, published in Sky Sports.
“I’ve come here to improve every day and to win titles with Real Madrid. These are my personal goals.
“I believe I have what it takes to play here. I’ve got everything I need and I want to enjoy football and to learn.
“I’m grateful to Tottenham, but I’m now opening a new page in my life,” he concluded.
By Gareth McKnight