The centre-back suffered a knee injury in the defeat to Chelsea on December 29 and then missed the win over Manchester United.
He is set to be sidelined for the rest of this season and may also miss the start of the next campaign.
It leaves Arteta short of options in defence and he will meet with technical director Edu and head of football Raul Sanllehi this week to mull over plans for the coming month.
Asked what fans can expect from Arsenal in the winter window, the Spaniard said: "That we will try to be in the market to see options that can strengthen the team, that's for sure and our obligation, and we will be working on that.
"My obligation is to give my opinion on the things we can improve. Obviously we have a bad injury with Calum a few days ago that will change our plans at the back."
Whilst confirming he will be looking to add to his squad, Arteta admits they will not be able to do much work on strengthening during the current month and will instead be trying to get injured players back on the pitch.
"I'm more concerned about getting players back from injury and try to improve the players I have here, and if something extra comes up and we think it is the right opportunity to improve what we have, then let's do it," he added.
Chambers' injury was described as "significant" by Arteta and, while he did not want to put a timescale on his recovery, he conceded "it is not looking good".
While Arsenal may now be more active in terms of recruitment this window, the Gunners head coach was firm on the future of captain Pierre-Emerick Aubameyang, who has only 18 months left on his contract.
"I don't even think about the possibility with Auba (leaving)," Arteta said. "I want him here."
The forward captained Arsenal to a 2-0 win over Manchester United on New Year's Day - their first victory at the Emirates since October 24 - and Arteta says that performance is the benchmark going forward for his players.