Arteta was injured in training ahead of Sunday's defeat to Manchester City and forced to miss his first Premier League game of the season.
A scan will determine the full extent of the problem, but Wenger does not expect Arteta back before the transfer window shuts.
The manager said: "He has a scan on Monday. He will be out for approximately three weeks. It is a calf strain."
Wenger confirmed he is seeking reinforcements as a matter of urgency as he prepares for a spell of five games in 17 days.
He said: "We are in the market. We try very hard (to strengthen)."