The Norway international moved to the Emirates Stadium on loan from Real Madrid in the January transfer window.
As things stand Odegaard is set to return to the Santiago Bernabeu at the end of the campaign.
However, the attacking midfielder faces an uncertain future after being sent out on loan four times since joining the Spanish giants.
Odegaard turns 23 later this year and may be keen to secure a permanent transfer away from Madrid if he feels first-team opportunities will be hard to come by again next term.
Arsenal would no doubt be interested in bringing the playmaker to the club on a permanent basis.
However, the Gunners would likely face stiff competition for the signature of a player with vast potential.
And while Arteta did not rule out a summer swoop for Odegaard, he says he will wait until the summer before discussing such matters.
“We have Martin for a few months from Real Madrid and first of all we have to maximise this time with him to try to bring to the team all the qualities he has,” the Arsenal boss said.
“At the end of the season we will sit together and decide where we can take that relationship forward. At the moment it doesn’t just depend on us.
“On the pitch he is showing a lot of character and a lot of intelligence. They are characteristics that a leader should have.
“He has been here a very short time but he has adapted very quickly. He is getting more and more respect from the players and the staff and his weight on the team will increase with every week.”
Arsenal return to action against Benfica in the Europa League on Thursday.
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