Bendtner netted Arsenal's opener in their 2-0 win over beleaguered Cardiff City on New Year's Day, but sprained his ankle in the act of scoring.
The Denmark international's absence comes as a blow to the Premier League leaders, who were without Olivier Giroud for the visit of the Welsh side due to an ankle problem.
And Wenger conceded that his side lack depth up front.
"It shows we are short," Wenger told BBC Sport. "He (Bendtner) sprained his ankle, (for) how long we don't know but it looks weeks."
Wenger professed that he would like to have more strikers available to him, but stopped short of expressing a desire to sign a centre forward in the January transfer window.
"We have Lukas Podolski and Theo Walcott who can play striker but over the longer period we want more specialists to play through the middle," he added.
"With two players injured it is difficult. Don't be too quick at the moment, we will just say we have two centre forwards injured."
