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The winger went down in the 10th minute of the Gunners' Champions League draw against Bayern Munich despite facing no challenge from the opposition.
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The Gunners star has had fitness issues in recent months and was managed at the end of last season prior to the World Cup.
During that time, he missed the March international break and seven games for the Premier League champions due to an Achilles injury.
Additionally, the 25-year-old only started three of England's eight games at last summer's tournament.
However, he remains vital to the Gunners, whose win percentage is notably higher with him in the team.
Ahead of the Gunners' fixture against Hull this weekend, Arteta said: "We are talking about weeks.
"Let's see, after the international break he will start hitting the pitch and doing more activities depending on how he feels.
"But unfortunately he is going to be out for a while."
He added: "I don't want to put too much pressure on the guys. They are doing the best to get him fit but it's still quite early."
Saka has made ten appearances in all competitions at club level this season, scoring four goals and assisting once
Three of his goals have come in the Premier League, while has scored and provided an assist in the Champions League.
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Meanwhile, Arteta has confirmed that Ezri Konsa and William Saliba are fit to face the Tigers this weekend.
Konsa has seemingly recovered from the knee issue he picked up in Wednesday's triumph over Slavia Prague.
Arteta said: "Yeah he's fine, he was feeling much better today, we took him out for precaution and for the amount of minutes that he was playing but yesterday and today he's feeling good."
The Spaniard is relaxed regarding Saliba's possible inclusion in Zinedine Zidane's second France squad.
Saliba is yet to play any minutes since returning from his back injury but he was on the bench midweek in the Champions League.
Arteta added: "Well, we've been in very close contact, I think they’re announcing it at 5pm this evening, so they have all the information.
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"Now it's about them, Zinedine in this case to make the call, and we will accept whatever it is."
The Gunners boss will be preparing for multiple players to leave the Emirates for international break next week, and will hope none will be returning with injuries.
The north London club face fellow title contender Manchester City and local rivals Tottenham in the next few weeks.