The Spaniard injured his calf during a UEFA Champions League group game with Borussia Dortmund in late November.
Wenger originally declared that Arteta would be out for a minimum of "two to three weeks" with the problem, and the Frenchman provided an update on the situation in a press conference on Tuesday.
"He's progressing well. He had a little setback and will not play before January," he said.
The news will come as a blow to Arsenal, who face the first of three Premier League outings in seven days when they host QPR on Boxing Day.
Wenger also confirmed that Aaron Ramsey will miss the QPR game, as well as the trip to West Ham two days later, but could be in line for a return at Southampton on New Year's Day.
Like Arteta, Germany international Mesut Ozil is also set to come back next month.
"He'll be back at the beginning of January," Wenger said.
"He works very hard and is very focused. Fitness wise, he's there."
The trio of goalkeeper David Ospina, midfielder Tomas Rosicky and forward Yaya Sanogo have all been passed fit for Friday's game at the Emirates Stadium, while Hector Bellerin and Laurent Koscielny could be in contention to face West Ham on Sunday.
