Real Madrid reportedly discussed the highly-rated 21-year-old during the most recent transfer window as a replacement for right-back Danilo, Dani Carvajal's former deputy, who was sold to Manchester City for a £26.5m ($34.8m) fee in July.
Odriozola remained at Anoeta, however, eventually signing a contract extension that ties him to his boyhood club until 2022.
The San Sebastian native has continued to go from strength-to-strength at Real Sociedad under the tutelage of Eusebio Sacristan, starting eight of the club's first nine matches across all competitions in 2017-18.
He also appears to be firmly in the running for a place in Spain's World Cup squad after making his senior international debut, registering an assist in a penultimate qualifying win over Albania last week.
It is possible that former winger Odriozola, who played in the third tier of Spanish football just 12 months ago and only made his first-team debut in January of this year, could be playing for Real Madrid by the time the planet's biggest single-sport competition rolls into Russia in 2018.
According to AS, the reigning Spanish and European champions are weighing up another approach for the player and will have no fears over paying his €40m release clause should a rapid rise continue along a similar trajectory.
Carvajal remains Los Blancos' first-choice right-back during his second stint in Madrid, although it has been suggested that he could miss a minimum of two months after being diagnosed with a viral pericardium infection.
Moroccan academy graduate and Castilla regular Achraf Hakimi, 18, made his debut in the recent 2-0 victory over Espanyol and will be expected to fill the void until Carvajal returns to full fitness. Spanish radio station Onda Cero now claims that the latter could resume first-team training as early as next week.
Real Madrid are not the only La Liga heavyweight to be linked with a potential swoop for Odriozola. During the early stages of the summer, it was reported that fierce rivals Barcelona were looking at him, Valencia's Joan Cancelo and Southampton's Cedric Soares as potential alternatives to first-choice right-back targets Hector Bellerin and Nelson Semedo.
Blaugrana opted to sign the latter at an initial cost of €30m, and the early indications are that they have now secured their long-term successor to Dani Alves. Despite that success, however, Mundo Deportivo recently revealed that Barcelona have not forgotten about Odriozola and are still receiving reports on both the Basque defender and teammate Mikel Oyarzabal.