Odriozola was part of Spain's 2018 FIFA World Cup squad and is due to sign a six-year deal with Los Blancos.
Odriozola is a promising young defender whose performances for Real Sociedad last season earned him a place at the World Cup. Spain have plenty of options in that position, and Odriozola made the cut ahead of more established players like Barcelona's Sergi Roberto and Arsenal's Hector Bellerin.
The full-back featured in the group games against Switzerland and Spain, although it was a disappointing tournament for La Roja as they suffered defeat to hosts Russia in the last 16.
The youngster will now head to the Santiago Bernabeu and will cost Real Madrid €40 million (£35.3 million), per Roberto Ramajo at AS.
Odriozola will add to Real Madrid's depth at right-back. Dani Carvajal is the club's first choice, while Nacho Fernandez can play at right-back or in central defence. The club also have 19-year-old Achraf Hakimi, who is expected to join Borussia Dortmund on a two-year loan, per Marca.
La Liga writer Andrew Gaffney offered his view on the signing:
The move sees Real Madrid continue to bring young Spanish players to the Santiago Bernabeu as they look to the future. Odriozola follows players such as Dani Ceballos, Jesus Vallejo and Marco Asensio in making the move to Real Madrid.
Freelance football writer Simon Harrison showed how the deal works out for Real Sociedad:
Odriozola is a pacy right-back who loves to attack and will hope he will be given opportunities to impress in the first team next season.
New Madrid boss Julen Lopetegui faces a tough job after replacing Zinedine Zidane. The Frenchman guided the team to three consecutive UEFA Champions League wins, but they finished in third place in La Liga last season and a distant 17 points behind fierce rivals Barcelona.