Odegaard hobbled off the pitch and was photographed using crutches as he boarded a plane back to London where he underwent further scans to reveal the extent of the injury.
Arsenal face north London rivals Tottenham Hotspur in the Premier League on Sunday when they will also be without midfielder Declan Rice who is suspended.
"Such ankle injuries often take at least three weeks. Anything other than that is just a bonus. And then it can take longer," medic Ola Sand was quoted as saying by Norway's VG.
"Ankle injuries are very painful right away. With Martin, it was extremely painful. He became very worried.
"What we have so far obtained from the MRI examination in London is that there is probably no fracture in the ankle. In the event of a fracture, we are talking about being out of action for six weeks plus."
Odegaard has become a talismanic figure at Arsenal and has missed only a handful of Premier League games in the last three seasons under Mikel Arteta.
A lengthy absence would be a huge blow to Arteta especially as new signing Mikel Merino also out with a fractured shoulder.
"His response to the rehabilitation in the next few weeks is the most important," Sand added. "It will start carefully on the bike, carefully in relation to what he can tolerate from pain. Then you take it gradually."
After the visit to Tottenham, Arsenal begin their Champions League campaign against Atalanta next Thursday.
