The 25-year-old has missed the Gunners' last three games after sustaining the problem on international duty with Norway on September 9.
News of Odegaard's imminent availability comes as a boost to the north London club, who were denied a statement win at champions Manchester City deep into stoppage time on Sunday, but Arteta insisted the Norway skipper is unlikely to be back before next month's international break.
Speaking ahead of Wednesday's Carabao Cup third-round tie against Bolton, the Spaniard said: "I think it will be a matter of weeks, but how long exactly I cannot tell.
"We might get a surprise but I think it's unlikely he'll be back before the international break."
Arteta said full-back Oleksandr Zinchenko would remain sidelined for "a few more weeks".
The Arsenal boss was also quizzed about his defensive tactics after the visitors went down to 10 men at the Etihad Stadium on Sunday, and said he would have been "very thick" if he had not learned from previous mistakes.
Leandro Trossard kicked the ball away and was sent off for a second bookable offence just before half-time, with the Gunners 2-1 up.
Arteta's side sat back and soaked up pressure for the vast majority of the second half before John Stones grabbed a dramatic equaliser for the hosts in the eighth minute of stoppage time.
"We had to play the game that we had to play," he said.
"The first 15 minutes, we couldn't. They played 30 seconds with 10 men (when Rodri went off injured). Look what they did (they defended). It's normal what they did."
Recalling Arsenal's heavy defeat at Manchester City in 2021, when former captain Granit Xhaka was dismissed, he continued: "Unfortunately we've been in the same situation. We were in that same situation with Granit and when we lost 5-0.
"So we'd better learn. If not I would be very thick."
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