The Gunners captain was sidelined with an injury while representing Norway during September's international break.
Odegaard returned to London on crutches wearing a protective boot and missed Sunday’s derby victory over Tottenham.
Arsenal had remained silent on the severity of the problem but Arteta has now offered a bleak update with his skipper facing a potentially long spell on the sidelines.
‘After recent scans, the scans showed that he’s got some damage especially on one of the ligaments in the ankle,’ Arteta told a press conference ahead of Thursday’s Champions League clash with Atalanta.
‘It’s something quite significant so we’re going to lose him for a while.’
Arteta was reluctant to give a precise timescale on how long Odegaard will be unavailable for, adding: ‘Hopefully not months, but let’s see.’
The Arsenal boss continued: ‘He’s our captain. He’s been one of the biggest and best players in our team.
‘Our identity is very linked to how he plays and behaves. It’s a big test for the team to see how able we are to show a different face.’
Arsenal also remain without Mikel Merino after he suffered a fractured shoulder in his first training session with the club.
There was good news on Arsenal’s other summer arrival Riccardo Calafiori however with the Italy international back in training and available for Thursday’s meeting with Atalanta.
Bukayo Saka also trained on Wednesday after he was forced off late on in Sunday’s 1-0 win at the Tottenham Hotspur Stadium.
It later emerged the England star was struggling with cramp and will be available on Thursday night – and more importantly, on Sunday for the trip to Manchester City.
‘Riccardo was able to train today and he will be in the squad,’ Arteta said. Saka was able to train today and completed most of the session so he’s available.