The Norwegian has been on loan at the Gunners since January, and has quickly become an important part of Mikel Arteta's side. Arsenal are hoping to strike an agreement with Los Blancos over Odegaard's future this summer.
However, according to the Daily Star, the north Londoners are concerned about getting into a bidding war with Liverpool and Chelsea, who are reportedly interested in the midfielder.
Zinedine Zidane's side reportedly value Odegaard at £30m but the report states Arsenal are afraid that the player's impressive performances could see the LaLiga giants increase their asking price to £50m.
Arteta's side will find it difficult to compete financially with Chelsea and Liverpool, and Paris Saint-Germain could also make a move for the 22-year-old. Odegaard has endeared himself to the Gunners faithful with his incredible goal in the north London derby win earlier this month.
Since making the switch during the winter transfer window, Odegaard has made 12 appearances across competitions, and was also on target in Arsenal's first-leg victory over Olympiacos in the Europa League last 16.
Odegaard hailed Arteta as "fantastic" and and admitted he was enjoying his time with the club so far. “Very good,” Odegaard said before adding, "He is incredible. I have a lot to learn from him. I just need to ‘sharpen my ears’ and pay attention. He is fantastic.”
He was also coy on his future at Real Madrid, saying: “I haven’t thought about what’ll happen in summer. Deal with Arsenal is until the end of the season. We’ll see what happens this summer. I’ve said things before that I still stand by: ‘Stability’ and ‘development’ are key words.”