Odegaard joined Sociedad on a one-season loan deal in July 2019, and has since shown exactly why Madrid went to so much trouble to snap up him up while he was a teenager at Stromsgodset.
The 21-year-old contributed an impressive seven goals and nine assists in 36 domestic outings for Imanol Alguacil's side this term, helping them earn a sixth-place finish in La Liga.
The Norwegian found the back of the net in the pulsating cup tie and set up another of the goals, before leaving the pitch to a standing ovation from the home supporters when substituted in the second half.
Sociedad legend Roberto Lopez Ufarte talked up Odegaard's abilities during an interview with Noticias de Gipuzkoa in April, while insisting the attacking midfielder could end up filling Luka Modic's boots at Madrid when the Croatian calls time on his career.
Ufarte said: “I think he has his head screwed on and he knows why he came here: to show people, to show that he has a lot of ability and that he is waiting for a spot to open up at Real Madrid. These are just my thoughts, but possibly in the next year when Modric leaves."
Madrid have yet to decide whether to recall Odegaard or extend his loan stay at Sociedad by an extra year, but Aperribay is "hopeful" an influential figure will stay put through to the end of the 2020-21 season.
Asked if there is a good chance of the former Stromsgodset starlet's services being retained, the La Real president told Onda Cero: "Yes, I think so. It doesn't depend on us but on him and Real Madrid, but we are hopeful that he can stay with us for another season."
Odegaard's final outgoing of the current campaign came as Sociedad earned a 1-1 draw with Atletico Madrid at Wanda Metropolitano on Sunday.
Madrid, meanwhile, became La Liga champions for the 34th time in their history after beating Villarreal last Thursday, before wrapping up their domestic schedule with a 2-2 draw at Leganes over the weekend.