Martin Odegaard could be farmed out on loan to gain more experience next season.
The 16-year-old made his debut for Real Madrid in the final game of the season, replacing hat-trick scorer Cristiano Ronaldo in the 7-3 victory over Getafe.
Despite his tender years, he has already made four full international appearances for Norway, and is tipped to go on and become a world star.
Spanish publication AS report Villareal, Real Betis and Malaga are all in the hunt for his services on a temporary basis.
After making his debut, Odegaard said: "It's a very special day for me and I'm very happy and proud.
"Of course it was very special to play in front of so many people. I tried to enjoy the moment and I did, so it was very good."
New Real boss Rafael Benitez could also look to offload 19-year-old Marco Asensio, another youngster on the periphery of the first-team squad.
