Modric passed his routine medical before being presented to a small gathering at the Santiago Bernabeu stadium where he signed a contract to conclude the deal that came at a reported cost of €35 million ($44 million) to Madrid.
"Finally this happened and I am relieved," Modric told a news conference. "It's the biggest challenge of my footballing career and I'm sure I will succeed. I have everything I need here, best coach in the world, best players, and I just need to enjoy everything and play my football."
Sunday's 2-1 loss at Getafe — which confirmed the club's worst start since 1973 — appears to have spurred Madrid into ending the drawn-out chase by securing a player it hopes can help kickstart its season. The Spanish giant did not release financial terms of the deal.
"It's not important how you start the league, it's important how you finish," said Modric, who added that he is ready to play Wednesday against Barcelona despite having only trained apart from Tottenham's team for the past month as the club negotiated his exit.
"There is no hard feelings from my side, and I think it's the same from Tottenham. This is just a new page in my life."
The 26-year-old Modric made 160 appearances for the north London club, scoring 17 goals, since joining in 2008 from Dinamo Zagreb for a fee of €21 million (then $32.7 million). Modric has scored eight goals from 57 appearances with the national team.
"Luka has been a terrific player for us, whilst we preferred not to part with him, we are pleased that it is to Real Madrid," said Spurs chairman Daniel Levy, who managed to keep Modric last season despite strong interest from Chelsea.
Modric's arrival also signals a possible exit for Madrid midfielder Kaka with only a few days remaining before the transfer window closes at midnight Aug. 31. AC Milan is reportedly interested in taking the Brazilian back to the San Siro.
Modric will likely train with his new teammates for the first time later Monday with a possible debut coming in the return leg of the Spanish Super Cup against Barcelona. Madrid trails 3-2 going into the game at the Bernabeu.
"Of course I would like to play, it's the biggest game in world football," Modric said. "The coach will decide where he wants to use me. I'm happy wherever he puts me."
Madrid has lost two straight games since opening the Spanish league season with a draw. It is already five points adrift of title rival Barcelona in the standings after just two league games.
