The Norwegian playmaker was a starter in his team's opening two league games but has seen limited action since.
Odegaard was not included in Zinedine Zidane's travelling squad for Wednesday's Copa del Rey game at Alcoyano (stream LIVE at 2:55 p.m. ET on ESPN+ in the U.S.).
His two-year loan at Real Sociedad was cut short last summer with Zidane keen to have Odegaard back in the fold as Madrid decided not to make signings due to financial hardship caused by the coronavirus pandemic.
Odegaard, 22, impressed with the Basque outfit, scoring seven goals and setting up nine assists in 36 appearances.
However, he has played 367 minutes in his nine games for Madrid since his return and believes the best move is to leave so as to avoid a career stall.
Sources have told ESPN that the club understands the player's desire and that most likely Odegaard will leave on loan before the transfer window ends.
Real Sociedad are monitoring the situation closely while Madrid sources have questioned if Zidane lacks patience with Odegaard or has simply been left unconvinced when the Norwegian has played.
Odegaard is one of many on a long list of footballers that were destined to achieve greatness at Madrid but have requested to leave due to a lack of opportunities.
Sergio Reguilon, who was one of the best left-backs in La Liga while on loan at Sevilla from Real Madrid last season, joined Jose Mourinho at Tottenham in September.
Achraf Hakimi is now shining at Inter Milan, despite an early struggle to adapt to Serie A.
Zidane did confront the criticism during Tuesday's news conference before it was known that Odegaard wanted to leave the club.
"It's easy to blame me," he said. "This is Real Madrid, it's complicated. There's a time when a player must play and prove himself. Then there are times that players have to leave to prove themselves. It's not the fault of the coach."
Serbian striker Luka Jovic had been the most recent example.
Jovic, 23, re-joined Eintracht Frankfurt last week on loan for the remainder of the campaign looking to spark his career just 15 months after arriving at Madrid in a reported €60 million transfer.
He scored two goals in the space of thirty minutes on his debut with Eintracht on Sunday, the same number of goals he had at Madrid in his spell at the club.
"I liked what he did," Zidane said. "It's what he knows. I'm happy. He's a Real Madrid player and I wish him the best.
"That's what Luka has. Luka is 23 and has a future ahead to prove what he is worth at Real Madrid. The same goes for everyone. There's strong competition here."
Sources have also struggled to understand the continuous benching of Fede Valverde, the lack of minutes of Mariano or why Brahim Diaz is starring at AC Milan -- the club he joined on loan ahead of the start of the season from Madrid.