Odegaard has missed Arsenal's last 10 competitive matches after he sustained an ankle injury while on international duty with Norway in September.
Earlier this month, Arsenal released footage of Odegaard stepping his recovery by working individually and running on an anti-gravity treadmill.
Norway boss Stale Solbakken recently revealed that everything is "going according to plan" for Odegaard to return to action in time for November's international window.
Arsenal handed Odegaard injury boost
According to the Daily Mail, Odegaard is now on the brink of taking the next key steps in his rehabilitation process.
The report claims that Arsenal's talismanic playmaker is set to return to training on the grass at some point this week.
While Odegaard is making progress in his recovery, it is still currently unclear when he will be ready to return to competitive action.
Odegaard will not be involved in Wednesday's EFL Cup clash with Preston North End and is unlikely to be ready for Saturday's Premier League meeting with Newcastle United.
The Gunners will then face Inter Milan and Chelsea, and Odegaard may have to play in at least one of those matches if he is to make himself available for international selection ahead of Norway's UEFA Nations League games with Slovenia and Kazakhstan.
Arsenal offered Gabriel, Timber hope
Arsenal were handed two injury concerns in Sunday's 2-2 draw against Liverpool after Gabriel and Timber both failed to finish the match at the Emirates Stadium.
However, the Gunners are believed to be hopeful that both players have avoided serious injury and will not spend too long on the sidelines.
Gabriel picked up a knee problem in the draw and is expected to undergo a scan on Tuesday to establish the full extent of the injury.
As for Timber, he complained of calf tightness after the match and will look to make a swift recovery as Arsenal prepare for four consecutive away games before the November international break.