Odegaard has made 14 appearances for Arsenal since joining in January – already more than the 11 he has made in total for Real in over six years since signing for Los Blancos aged 16.
Speaking earlier this month, Arsenal boss Mikel Arteta insisted Odegaard was happy at the club but did not discuss the possibility of him staying beyond this season.
“I speak with him every day,” Arteta said. “I see his face, his body language and he seems really happy right now here.
“Our contract with him is until the end of the season. What we have to do is try to make him perform, develop and be as happy as he can be.”