Odriozola officially joined Los Blancos earlier this month, and will join up with Lopetegui once again following their time together with the Spanish national team.
Lopetegui handed the 22-year-old his first Spain cap, and also named the defender in his La Roja squad for the 2018 World Cup before being sacked on the eve of the tournament.
Odriozola, who will seemingly battle Dani Carvajal for the right-back spot at the Bernabeu, has said that he 'really likes' Lopetegui as a coach.
"I have spoken with Julen Lopetegui and he has explained about having to give everything you have and how Madrid will have incredible demands," Odriozola told reporters at his unveiling.
"I really like him as a coach, he is a specialist at transmitting messages to the squad and he knows football inside out, he will be great for Madrid."
Odriozola could make his Madrid debut in their pre-season opener against Manchester United on August 1.