Odegaard hobbled off during his country's Nations League defeat of Austria in Oslo on Sept. 9 and returned to London on crutches.
He missed Sunday's North London derby against Tottenham Hotspur and Arteta provided an update ahead of Arsenal's opening Champions League game against Atalanta in Bergamo on Thursday.
"The scans showed that he's got some damage, especially on one of the ligaments in the ankle," said the Spaniard.
"It's something quite significant so we're going to lose him for a while."
Arteta, whose side are second in the league and two points behind Manchester City, said he hoped it would not be months "but let's see."
Odegaard has become a key player for Arsenal, missing only a handful of games over the last three seasons, and a lengthy absence will come as a huge blow.
"He’s our captain. He’s been one of the biggest and best players in our team," continued Arteta.
"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."
Norway coach Stale Solbakken told VG newspaper he had been fully updated about the injury and was "preparing for a life without Martin."
"We have to take it week by week and day by day. Some types of damage go faster and some go slower," he said.
