The Norway midfielder has just completed the first of a two season loan with La Real.
However, AS says Real Madrid coach Zinedine Zidane wants to work with Odegaard next term and La Real have already been informed of their plans.
Zidane was prepared to allow Odegaard to remain in San Sebastian, but after talks with the youngster he was convinced the midfielder was ready to compete for a place in Madrid's XI.
And so Real president Florentino Perez has informed his La Real counterpart Jokin Aperribay that Odegaard will not be returning next season.
Odegaard will miss the start of Real Madrid's preseason as he recovers from a knee injury suffered towards the end of last term.