That is according to club legend Nigel Winterburn.
He believes Odegaard may only want to play for the Gunners permanently if manager Mikel Arteta is still in charge next term.
Arteta's future is in doubt given Arsenal will be without any European football next season.
"Arsenal would be in prime position; he's been here he knows what the club is like. He probably wants assurances that Arteta is going to be in charge because he wouldn't want to sign for a different manager.
"He's usually in the team and he's shown enough. He's a link up player, he's very neat and tidy on the ball and we need creative players in around that area.
"I would think there will be other clubs looking at him as well, if he becomes available."
Winterburn added: "I think it comes down to when he's been at the club, how he feels, how he's been treated, and does he feel again that potentially it's somewhere where he wants to be?
"It's looking as if he would get quite a lot of game time, but is that enough for him?
"We've seen glimpses of what he can do, so that's an option I would think Arsenal would be looking at. It depends on the fee and whether Arsenal are willing to pay that fee, it all comes into the equation."