Spanish publication 'AS' reports that Odriozola is Real boss Zinedine Zidane's no. 1 target to strengthen his squad at right-back.
Whilst the performances of youngster Achraf Hakimi have largely been to Zidane's liking, 'Los Blancos' are thought to be planning on loaning the Moroccan out for a season or two so that he can gain regular top-level football.
That approach has worked wonders for the likes of Marcos Llorente and Marco Asensio and Real believe that it would really help Achraf develop as a player.
Whilst talks are ongoing over Kepa Arrizabalaga's move to the Bernabeu, Real bigwigs are also thought to be confident of persuading Sociedad to part with Odriozola for less than his €40m release clause.
Despite only turning 22 this month, the defender has already made his debut for the Spanish national side and is tipped to be included in their squad for next summer's World Cup.
With that in mind, Real are keen to complete a deal before the tournament, to avoid the youngster piquing the interest of other clubs.