The Germany international pulled a muscle during his side's 2-0 Champions League playoff victory over Fenerbahce on Tuesday.
"It is bad news about Lukas Podolski as he will be out for eight to 10 weeks," Wenger told the club's website ahead of Sunday's north London derby against Tottenham Hotspur.
"His hamstring is much more damaged than we thought at the start. It is very disappointing and we have paid a very heavy price for that win. We have to be patient now."
The 28-year-old had started the season strongly, scoring scored in Arsenal's first two Premier League matches against Aston Villa and Fulham.
Wenger said Aaron Ramsey, who scored both goals against Fenerbahce, and Jack Wilshere would be fit to play against Spurs having both suffered injury scares in the match against Fenerbahce.
Midfielder Mikel Arteta is still a few weeks away from returning but long-term absentee Thomas Vermaelen could be back after next week's international break.
(Reporting by Josh Reich, editing by Ed Osmond)
