The 23-year-old, who was making his first start after returning from a calf injury, went down clutching his knee while challenging for the ball in the second half and was carried off the pitch on a stretcher.
His injury is another blow to manager Unai Emery, who had already lost defender Rob Holding for the rest of the season to a similar knee problem last month.
The Spanish coach said he had players to fill in for Bellerin and was unlikely to dip into the transfer market to sign a replacement, having previously said he could only take players on loan in the January transfer window.
"We have Ainsley (Maitland-Niles). He can help us by playing like a right back or a right winger. And also, (Stephan) Lichtsteiner is another player," Emery said.
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"I don't forget (Carl) Jenkinson... he played some matches and if we need (him), he is okay to help us."
Arsenal next host Manchester United in an FA Cup fourth round tie on Friday, before returning to Premier League action against Cardiff City at home on Jan. 29.
(Reporting by Rohith Nair in Bengaluru; Editing by John O'Brien and Christian Radnedge)
