The 23-year-old has started just three games across all competitions this season but his absence tests the Gunners' depth in midfield with Thomas Partey and Emile Smith Rowe also sidelined until 2024.
Speaking on Tuesday ahead of Arsenal's Champions League home clash against French side Lens, manager Mikel Arteta said: "We've got a problem with him [Vieira]. He's been having some discomfort. We tried to have some treatment with him but he wasn't improving.
"We sent him to see a specialist and the advice was to get an operation done in his groin. He did that yesterday. We expect him to be out for weeks now. We don't expect him back this side of the winter break. Let's see how he evolves. We will have to wait and see."
Arsenal are already eyeing midfield options in the January transfer window with Aston Villa's Douglas Luiz the club's preferred target, while Arteta admitted Jurriën Timber's long-term knee injury has left the Gunners short in defence.
"We have been very unlucky with some of the injuries," Arteta continued.
"Some others happen. They happen to any other club. Where they happened, it is a bit critical. The one with Jurriën as well because we were really light in the defensive line, that was a huge blow. But the team has to find ways to perform and adapt to win games."
Arsenal can confirm their status as Group B winners with victory over Lens but will need to avenge a 2-1 defeat in October's reverse fixture.
And ahead of the five-year anniversary of Saka's senior Arsenal debut, Arteta said: "He is targeted because he is one of our most dangerous players. The teams know that and they try to stop our strengths. What he has done in those give years I think is remarkable for his age.
"When you look at his stats, how consistent he has been, it is very rare to see that at his age especially in those positions at top clubs so big compliment to him. He is a top, top player that's for sure.
"Getting to the top is one thing, maintaining it is something else. He has done it for five years. You can see the way he is improving is really steady but really fast as well. He has the right mentality because he has the right family, the right people around him.
"He has the right advice around him and that's what makes him special. He wants much more because there is much more to come."
Arteta also confirmed Ben White has recovered from illness sufficiently to be in contention to start against Lens.