The 22-year-old spent the latter part of 2020/21 with the Gunners and, while he only made nine Premier League starts, his impact in a playmaking role was plain to see.
According to the Daily Star, Real are looking to sell fringe players in order to boost their transfer budget - and, while new boss Carlo Ancelotti could include the Norway international in his plans, Odegaard, ostensibly, falls into that bracket.
Mikel Arteta is a huge admirer of Odegaard and would no doubt jump at the chance to bring him to North London on a permanent basis, especially with Arsenal lacking in terms of creativity.
Emile Smith Rowe has emerged as a key man for Arteta's side, but the Gunners can't be overreliant on the 20-year-old, despite his prodigious talent. Odegaard would nicely complement the player who a certain commentator insists on comparing to a certain Manchester City midfielder by giving him a certain nickname which hasn't exactly caught on.
Odegaard made 20 appearances for Arsenal in the Premier League and Europa League, scoring twice - most notably in March's 2-1 North London derby win over Tottenham at the Emirates.
He joined Real Madrid from Norwegian club Stromsgodset in January 2015 but has only made 11 first team appearances for Los Blancos, also spending time on loan at Heerenveen and Vitesse Arnhem in the Netherlands, and Real Sociedad.
Subscribe to FourFourTwo today and get a free England Euro 96 shirt!
RICHARD JOLLY Can England become the new Germany at major tournaments?
TRANSFER NEWS Chelsea preparing huge offer for Erling Haaland
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.