The Norway international has missed the Gunners' last 11 matches in all tournaments due to his severe sprain, which he picked up in his side's Nations League clash with Austria in September.
During Odegaard's absence, Arsenal have only won three of their six Premier League matches to fall five points behind Manchester City in the top-flight table, while also sitting four points worse off than Liverpool.
It was reported earlier this month that the former Real Madrid starlet had suffered a setback in his recovery and could be missing until December, but Mikel Arteta quickly allayed those fears.
Now, it appears that talk of a Christmas comeback was wide of the mark, as the Evening Standard reports that Odegaard is targeting a return for next week's trip to San Siro in the Champions League.
Odegaard has 'outside chance' of making Newcastle United squad
Before Arteta's men face the reigning Serie A champions, they will pit their wits against Newcastle United in the Premier League on Saturday, and Odegaard apparently has an 'outside chance' of being fit for that clash as well.
However, a comeback against Inter Milan is more realistic for the 25-year-old, who has been making great strides in his recovery and is 'itching' to be back on the field again.
The Gunners are not prepared to take any risks over the fitness of their skipper, though, especially as their first Premier League game after the trip to San Siro comes away to Chelsea on November 10.
The report adds that Odegaard has now entered the last stage of his rehabilitation, and a return date will be confirmed after Gunners medical staff assess how well he comes through that final step.
While Odegaard might be involved at St James' Park this weekend, Riccardo Calafiori (knee), Kieran Tierney (hamstring) and Takehiro Tomiyasu (knee) are still sidelined, while Gabriel Magalhaes is a doubt with the knee issue he sustained against Liverpool.
How big a boost will Odegaard's return be for Arsenal?
A quick glance at the goalscoring metrics would suggest that Arsenal have not missed Odegaard too much, as the Gunners have scored a total of 23 goals in their 11 games without him, an average of just over two per game.
However, their lack of ingenuity without their captain has come to the fore in a couple of their matches, notably their 0-0 Champions League draw with Atalanta BC and 2-0 loss to Bournemouth, where they were found wanting even before William Saliba's red card.
Bukayo Saka's creative prowess has negated the impact of Odegaard's absence, although that has come at a slight cost to his goalscoring numbers, and the return of the Norwegian should allow Arsenal to start controlling games again.
Gunners boss Arteta has been criticised for adopting a Jose Mourinho-like approach to games, putting men behind the ball with a one-goal lead as was the case against Liverpool, and one cannot help but wonder whether Arsenal would have tried to kill the game off had Odegaard been available.
Ethan Nwaneri's wonder strike against Preston North End proves that Arsenal could still be in good hands without Odegaard, but the 17-year-old is not yet being considered for Premier League starts, so having the skipper back in the ranks is a wonderful piece of news for Arteta.