Modric flew into Madrid on Monday morning and after passing a medical signed a five-year contract in the presence of big-spending Real president Florentino Perez before holding a news conference at the Bernabeu stadium.
The 26-year-old, who will wear the number 19 shirt, said he was relieved the protracted transfer negotiations had concluded before the transfer window closes on Friday and he was prepared to fight to win a place in Jose Mourinho's team.
"I have to work very hard to earn a spot in the starting XI and I'm prepared to do it," he told reporters as his wife and young son looked on.
"I've come here to improve every day and to win titles with Real Madrid. These are my personal goals," he added.
"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."
While neither club gave financial details, Spanish media reported Real would pay 35 million euros for Modric, plus another seven million depending on the number of his appearances and the team's performance.
His arrival in Madrid may help Real put a poor start to the season behind them, during which they have taken one point from two La Liga outings and lost Thursday's first leg of the Spanish Super Cup 3-2 at Barcelona.
They crashed to a shock 2-1 defeat at city rivals Getafe on Sunday, which prompted a rare burst of public criticism of his underperforming players from a furious Mourinho.
Real have a chance to reignite their season in Wednesday's Super Cup return leg at the Bernabeu when Modric hopes to make his debut.
"Real against Barca is one of the biggest games in world football and I would love to play but the coach will decide," he said.
FAN FAVOURITE
Spurs chairman Daniel Levy, who fought off bids from Chelsea and Manchester United for the fan favourite a year ago, said he was reluctant to let Modric leave.
Modric himself said last year he wanted to move to Chelsea but Levy blocked any deal with the club's London neighbours and Modric buckled down to have a fine season with Spurs who finished fourth in the Premier League.
"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," Levy said in a statement.
Modric joined Tottenham from Dinamo Zagreb in 2008 for 16.5 million pounds and after initially playing as a defensive midfielder, became a key element in the club's resurgence under former manager Harry Redknapp.
After a patchy start to that first season under Juande Ramos, the Spaniard's replacement Redknapp deployed Modric as an attacking, creative midfielder and Spurs rapidly improved.
With Redknapp at the helm, Spurs enjoyed two top-four finishes and reached the Champions League for the first time in 2010 when Modric's deft touches and vision set him apart.
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