The 21-year-old midfielder was unveiled at the Santiago Bernabeu after passing a medical and putting pen to paper on a six-year contract with the La Liga giants.
Kovacic also reportedly had the chance to Liverpool, but the Croatian admits his decision was a no-brainer when Real showed an interest.
He said: "Real Madrid is the biggest club in the world, I have no doubt about it.
This opportunity comes once in a lifetime
Mateo Kovacic
"This opportunity only comes once in a lifetime.
"When they told me about Real's offer, you don't have any doubts.
"I am young, I have a lot to learn and I hope I can give the best of me.
"To be able to share the changing room with such great players, it's truly an honour."
GETTY
Mateo Kovacic thrilled with Real Madrid move
Kovacic spent two seasons at Inter after joining from Dinamo Zagreb in the summer of 2013 and will play alongside compatriot Luka Modric at Real.
He added: "Luka ia a great friend and I'm very happy I will be able to play with him.
"Real have a great team and I'm looking forward to playing with Cristiano Ronaldo."
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