The 27-year-old midfielder has just two years left on his contract, which runs until June 2028, and has been hit by persistent injury problems.
Odegaard made 36 appearances in all competitions last season but missed a significant number of matches through injury and lost his place in the starting eleven.
His form has also dipped in recent times, contributing to Arsenal's willingness to sanction a sale, says BBC Sport.
Turkish journalist Selman Öztürk reports that Odegaard has already reached personal terms with Galatasaray.
Björn Kvarme, the player's agent, travelled to Istanbul last week to help finalise the move, which Odegaard has reportedly approved.
Turkish clubs are investing heavily in this summer's market as they look to grow their standing in European football, and formal negotiations with Arsenal are expected in the coming weeks.
Galatasaray hope to complete the deal for around €40million, according to the report.
Arsenal are also facing the departure of Leandro Trossard, who has been at the club since January 2023.
The Belgian scored eight goals and provided ten assists in 50 competitive matches last season and is under contract until the summer of 2027.
Turkish journalist Yağız Sabuncuoğlu now reports that Besiktas have submitted a bid for Trossard worth €18million plus €2million in bonuses.
Arsenal therefore face the prospect of losing two experienced attacking players before the new season gets underway.