Per journalist Ben Jacobs, the club believe they will secure the Norway star's signature very soon.
Odegaard has been a revelation since arriving at the club, initially on loan, from Real Madrid.
Speaking to GiveMeSport, Jacobs said: "If you've brought in someone like Declan Rice, you might have broken your wage structure a little bit.
“Someone like Odegaard, on £115,000-a-week, might be wanting up to £250,000-a-week - you just don't know because a lot can change when your project grows very fast.
"But there's optimism. and there is no reason to think that Odegaard won't sign. I would expect, now the international break is over, for talks to continue and move relatively fast towards some kind of agreement in principle.
Jacobs later added: "Arsenal are very optimistic that Odegaard will be another key member of the club to renew."