The Norwegian playmaker joined Arsenal on a six-month loan during the January transfer window.
But even if the Frenchman does depart, Spanish outlet Sport says Real could look at selling Odegaard.
The Spanish giants want a ‘good offer’ to consider transferring the 22-year-old.
It is unclear what would constitute an acceptable offer for Los Blancos.
Regardless, Odegaard must impress Arsenal over the next four months if he wants the club to part with significant money to secure his signature.