Modric arrives on a five-year deal, and will wear the No.19 shirt at Santiago Bernabeu, as the No.14 he wore at Spurs is already taken by Xabi Alonso.
The Croatia international played no part in the north Londoners' Premier League campaign this year as the two clubs finalised negotiations for the deal, which also includes a long-term 'footballing partnership'.
|22/1||Norwich City are 22/1 with Bet365 to beat Tottenham 2-1 on Saturday
"I look forward to playing with the team. This is my biggest challenge as a footballer and I am sure that I will accomplish my objectives.
"My challenge is to earn a place in the team, and for that you must work hard. It's going to be difficult, but I'm ready. My challenge is to win titles and trophies with Real Madrid.
"I have trained, I have worked hard and I have played 25 minutes with Croatia. I hope that I only need a couple of games to be fit and to reach the highest level.
"I have everything that I want. Now i just want to enjoy my football, improve my qualities and help the team win everything.
"I'm grateful to Tottenham. I spent four fantastic years there. There is no bitterness, just gratitude."