Modric, speaking at a press conference in Madrid to announce his move, told a smattering of journalists: "I'm ready for everything. I'm going to work hard for the team, and hopefully my qualities can bring success to the team.
"I am proud and honoured to be here. I have trained and worked hard since Euro 2012. After a couple of games, I hope to be at my top-level form.
"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.
"It's an honour to play for Real Madrid. It's the best and the world's greatest football club."
Real confirmed the transfer on their website after the Croatian playmaker passed a medical. "Real Madrid CF and Tottenham Hotspur have reached an agreement for the transfer of the player Luka Modric, who will remain tied to the club for the next five seasons," the statement read.
The transfer is estimated to be around £35m, including add-ons. Modric becomes Real's first signing of the close season.
An earlier statement ahead of the medical included news of a partnership agreement for the two clubs to work together "in respect of players, coaching, best practices and commercial relationships".
The 26-year-old's arrival in the Spanish capital 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 coach Jose Mourinho.
Real have a chance to get their season back on track in Wednesday's Super Cup return leg at the Bernabeu when Modric, who will be presented to the media later on Monday, could make his debut.
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.
"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 16.5m and became a key element in the club's resurgence under former manager Harry Redknapp.
After a poor start to the season under Juande Ramos, the Spaniard's replacement Redknapp deployed Modric as attacking 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 inspired vision set him apart.
In all, he played 160 times for Spurs, scoring 17 goals, and was the creative heart of the side, whose attacking, expansive style was widely regarded as being among the most attractive in the Premier League last season.
Despite finishing fourth last season, Spurs were deprived of a Champions League berth after sixth-placed Chelsea earned the right to defend their title when they became European champions in May.
With Spurs only able to offer the Croatian Europa League football, their chances of keeping Modric at White Hart Lane were diminished.
His move to Madrid could now spell the end of Kaka's stint at Real, the Brazilian playmaker who has failed to live up to expectations since arriving from AC Milan three years ago.
Milan wanted to take the 30-year-old back on loan, although Real were holding out for a sale, the Italian side's vice president Adriano Galliano said on Sunday.
- Luka Modric
- Real Madrid