Chilean Sanchez, who sustained the initial injury against Norwich City at the end of November, was thought to be in with a chance of returning for Monday's home clash against Manchester City but was not on the bench.
Arsenal won the match 2-1 to move four points clear of City and stay second in the table, two points behind surprise leaders Leicester City.
"I think he will be back around the 10th of January. For the Christmas period, don't count him," Wenger told reporters, adding that Sanchez had suffered a "very slight" setback in training on Saturday.
"He was planned to be on the bench tonight but he is so keen that he got a little pain and we didn't take a gamble at all."
Arsenal travel to face Southampton on Dec. 26 before hosting Bournemouth two days later. They then face Newcastle United on Jan. 2 before taking on Sunderland in the FA Cup third round.
(Reporting by Martyn Herman; Editing by Peter Rutherford)
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