Odegaard twisted his ankle in Norway’s Nations League match against Austria earlier this month.
His national team doctor had suggested he would be sidelined for a “minimum of three weeks”.
“After everything was scanned, they showed that there was some damage, especially in one of the ligaments in the ankle so we’re going to miss him,” Arteta told reporters ahead of Arsenal’s Champions League opener against Atalanta on Thursday.
Arsenal, Premier League runners-up last season, visit reigning champions Manchester City on Sunday.
The Gunners beat local rivals Tottenham 1-0 last weekend to sit second in the early Premier League table, two points behind Man City.
