Having enjoyed a successful loan spell with Real Sociedad last season, he has been limited to just nine games under Zinedine Zidane this year.
With the Norway international looking to get first-team football, it has seen the north London side become a viable option for the Bernabeu side to offload the youngster.
The attacking midfielder has only played five minutes of football sine his start in the 2-0 Champions League loss to Shakhtar Donetsk.
And the report states that the player currently “feels discomfort in his knee”, meaning he would need treatment on his arrival at the Emirates Stadium.
Odegaard burst onto the scene at 15 years old in his debut season for Norwegian side Stromsgodset six years ago.
Scoring five goals in 23 games, he caught Madrid’s eye and earned a €3million move to the Bernabeu.
After working his way up the ranks at the Castilla Academy but was unable to get regular game time under Zidane.
Loan moves to Heerenveen, Vitesse and Sociedad have followed for the last three seasons before a senior squad role this year.
He scored seven goals on loan with the La Liga side - and even scored against his parent club to knock them out of the Copa del Rey.
While Madrid appear to be no longer interested in the 22-year-old, he has been the subject of a charm offensive by Arsenal director Edu.