The 26-year-old has spoken for the first time as a Blancos player, reiterating his desire to challenge at the very top despite missing out on the entirety of pre-season
The successful resolution to negotiations brings to an end one of the most protracted transfer sagas of the summer transfer window, with the constant haggling over a transfer price between the two clubs threatening to scupper the switch.
Modric had agreed personal terms with the Santiago Bernabeu outfit beforehand, but Spurs chairman Daniel Levy's financial demands initially proved problematic, a circumstance the 26-year-old has now acknowledged with regret.
"It [negotiations] has been a really long process, and I've missed two games," uttered the Croatia international at his official unveiling.
"I have been training with Tottenham’s fitness coach, working hard in addition to playing for Croatia. I hope I will only need a couple of games to get fit and reach my peak fitness level and that expected of me by Real Madrid."
The creative midfielder then spoke of his enthusiasm regarding the prospect of playing under coach Jose Mourinho, insisting he is up to the challenge of playing on the greater stage.
"I'm happy, honoured and proud to be at the biggest club in the world," Modric enthused. "When I realised that Real Madrid wanted to sign me, I was excited that such a big club were showing an interest in me."
"I want to play football and get a place in the team. At Tottenham, I played in the centre of the park and we’ll see how Mourinho, who is the best coach in the world, wants to use me.
"It will be difficult but I think I have the quality to play here. I have set myself the challenge of winning trophies with Real Madrid."
Indeed, Mourinho will be hopeful his latest signing will spur on the Blancos in the club's already-stuttering title defence, after seeing his men only collect a single point in their opening two Liga fixtures.