The former Barcelona star sustained his injury against Norwich last month but was given a slim chance of being fit enough to take some part in his side's top of the table clash against Manchester City on Monday evening, but was not even named in the matchday squad.
"Sanchez I think will be back around the 10th of January. He has had a slight setback two days ago. We had planned to put him on the bench tonight," Wenger said following his side's 2-1 victory over Manuel Pellegrini's side.
Arsenal cemented their place as Leicester City's nearest challengers with an impressive win over City which came courtesy of goals from Theo Walcott and Olivier Giroud.
Wenger's side will be without Alexis, however, for the Boxing Day trip to Southampton as well as upcoming home games against Bournemouth and Newcastle.
But last season's top goalscorer could be back in time for the game against Sunderland in the third round of the FA Cup, as the Gunners begin the defence of the trophy that have won two years in a row.