The Norway midfielder has already missed 13 games so far this season due to injury.
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Odegaard picked up a knock in the draw at Brentford in midweek and was seen limping after the match at the Gtech.
The Mirror says he faces another spell on the sidelines due to the knock.
The Gunners host Wigan today in the Cup before going to Wolves on Wednesday to resume their Premier League title campaign.