Odegaard, 22, wants to go out on loan for the second half of the season for more playing time, and Real have accepted his demands.
Real Sociedad, where Odegaard played last season, are interested in signing him - but Standard Sport understands Arsenal are in talks with Los Blancos and are growing in confidence that he may join them instead.
Following Arsenal’s FA Cup exit at Southampton on Saturday, Arteta was asked whether the Gunners were close to sealing a deal for Odegaard.
Arteta replied: “I don't know. Obviously we played today, so since last night I don't have any news.
"Probably, I will get a few phone calls today and see where we are with every deal.”
While Arsenal struggled for creativity up until recently with Mesut Ozil banished from Arteta’s Premier League and Europa League squads, one positive has been the emergence of Emile Smith Rowe.
The Arsenal academy graduate has quickly become a key player for the Gunners, though there is a concern that Odegaard’s arrival could mean less game time for Smith Rowe.
Asked whether Odegaard’s potential arrival could hamper Smith Rowe’s development, Arteta remained tight-lipped - but seemed to suggest that it will be up to each and every player to take their chances to develop.
"I cannot talk about any player who is not ours. We have a very clear idea of how we want our players to develop and what they need to do that. All the decision we make are just to reinforce their development and try to give them the best possible chance."