The 22-year-old has offered Mikel Arteta a valuable option in attacking midfield since his arrival on loan from Real Madrid during the January transfer window.
Since joining Los Blancos as a 16-year-old in 2015, Odegaard has been out on loan four times, excelling at Real Sociedad last season.
Standout performances in the Spanish top flight did not guarantee him regular opportunities back at Real this season however with Arsenal now benefitting.
Reports in Spain since his arrival in north London suggest Real still consider the Norwegian a big part of their future. Odegaard suggests however he will need guarantees over opportunities at the club before he makes his final decision.
‘I said many times that I have been looking for stability [at the same club], but haven’t found it,’ Odegaard told VG.
‘I think it will be important to find stability. I am very happy at Arsenal, enjoying myself. I like it here.
‘Now my focus is to finish the season then we will see.’
Gunners boss Mikel Arteta suggested in February the club will explore the possibility of extending his stay beyond this season.
“We have Martin for a few months and first of all we have to maximise this time with him to try to bring to the team all the qualities he has,’ Arteta said.
‘At the end of the season we will sit together and decide where we can take that relationship forward.’