Odriozola has left Real Sociedad to compete with Dani Carvajal for the right-back spot with the Spanish giants, and the new arrival has admitted that he is fulfilling his childhood dream.
At his official unveiling, the 22-year-old told reporters: "It is an honour for me to be here and fulfil the dream of every football player to sign for Real Madrid.
"I know the responsibility that comes with wearing this shirt and the only thing I can promise is that I will give everything, humility and sacrifice so that our future is successful.
"The last year and a half has been like a movie and I'm now fulfilling the dream of wearing the shirt of the biggest club in the world, that of Real Madrid. It's a day I'll never forget and I'll not let this opportunity pass me by."
Odriozola joins Real having made a total of 57 appearances for his previous club, while he has already earned four caps playing for Spain.