Tottenham are reported to have received £33million for the Croatia playmaker, who passed a medical at the Bernabeu on Monday morning.
"I'm very happy, honoured and proud to be at the biggest and best club in the world. I'm very excited," Modric said.
Tottenham are losing one of the Barclays Premier League's most creative players, who had also been linked with Manchester United and Chelsea during the past year.
Modric joined Tottenham from Dinamo Zagreb for £16.5million in 2008 and was highly influential under former manager Harry Redknapp.
With Modric pulling the strings, Spurs secured two top-four finishes in the Barclays Premier League and qualified for the Champions League for the first time in 2010.
The 26-year-old attacking midfielder also impressed during their European campaign, alerting Chelsea and United who were unsuccessful in attempts to sign him.
However, Spurs' failure to qualify for this season's Champions League and the interest of Real Madrid meant his time at White Hart Lane was over.
He made 160 appearances for Spurs, scoring 17 goals, and his transfer comes alongside a new partnership between Tottenham and Real.
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Spurs chairman Daniel Levy was reluctant to let Modric, who will wear the number 19 shirt at the Bernabeu, leave.
"Luka has been a terrific player for us and while we preferred not to part with him, we are pleased that it is to Real Madrid, a club with which we now look forward to sharing a long and productive partnership," said Levy.
Modric, who was left out of Tottenham's squad for the opening two games of the season, was determined to leave White Hart Lane as soon as he heard of Real's interest.
"I chose the Spanish league because it's one of the best leagues in the world," he said at a press conference at the Bernabeu.
"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.
"This is the best club in the world, with the best coach, the best players.
"Cristiano Ronaldo, Karim Benzema, Angel Di Maria, Gonzalo Higuain, Mesut Ozil, Xabi Alonso - I look forward to playing with them.
"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 have 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," he 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 manager Andre Villas-Boas to strengthen his squad.
Spurs' are reported to be interested in Brazilians Willian of Shakhtar Donetsk and Internacional's Leandro Damiao, as well as Fulham playmaker Mousa Dembele.