The Norwegian spent the second-half of last season on loan at Arsenal.
Mikel Arteta was impressed by Odegaard's performances and wanted to sign him on a permanent deal.
But those hopes have been dashed after the 22-year-old stated his intent to return to Real.
Speaking to Norwegian outlet TV2, the 22-year-old said: "Real Madrid have been clear that they would like me back.
"So then it will be time to go there when the training starts up again."
Asked about his aspirations to play for Los Blancos, Odegaard added: "Of course it has always been a dream to play there.
"I have been in the club for over six years now, and that has always been the goal.
"I always want to play, I have said that all the time. Playing hard is important."