The Norwegian joined the Gunners on a short-term loan in January after playing under 400 minutes of first-team football for Los Blancos during the first half of the campaign.
He has since impressed by scoring a couple of times for the north London side and has also played in the build-up of other goals with his quality distribution in the final third.
Los Blancos reportedly valued the 22-year-old at around £30 million in January, but it is claimed that his price tag could rise to £50m, should he continue to excel for the Gunners.
In such a case, Mikel Arteta’s side may not be able to afford his services, and it is reported that Chelsea and Liverpool could have an advantage in the pursuit of Odegaard.
The Blues are loaded with plenty of attacking options in their squad, and they brought in the likes of Timo Werner, Kai Havertz and Hakim Ziyech last summer.
However, manager Thomas Tuchel will want signings of his own, and Odegaard could be an option for the German if he is eyeing another creative midfielder.
Tuchel has regularly played with three at the back, but he may revert to his preferred 4-2-2-2 formation. Odegaard could be an option to play behind the front two.
Meanwhile, the Reds are likely to require reinforcements in the attack this summer after a poor defence of their Premier League title this season.
Jurgen Klopp has generally preferred a 4-3-3 setup, but it remains to be seen where he plans to switch his tactics to facilitate the signing of a number 10.
While Mohamed Salah, Sadio Mane, Roberto Firmino and Diogo Jota have been among the goals, there have been limited chances created behind them.
Odegaard has shown from his short time in the Premier League that he has a keen eye for the final pass and should only get better with more experience in the division.
Stats from Transfermarkt.com