The German international is expected to be ruled out for at least a month after picking up a groin injury during pre-season.
Odegaard has been strongly linked with a summer move, but according to AS the 22-year-old, as well as Spaniard Dani Ceballos, are both set to stay at the Santiago Bernabeu.
Both midfielders were on loan with Arsenal last season, but Mikel Arteta seemingly has more of an interest in luring the Norwegian back to the Emirates.
Odegaard made 20 appearances for the Gunners across all competitions, scoring two goals and providing two assists.
If Arteta does fail in his attempts to sign Odegaard, then Arsenal may step up their pursuit for Leicester City's James Maddison, with the north London club reportedly preparing a player-plus-cash offer for the 24-year-old.