The Germany international, who scored twice in the 3-1 Barclays Premier League win at Fulham last weekend, hobbled off during a 2-0 defeat of Fenerbahce as the Gunners progressed to the Champions League group stage draw.
Tests have revealed Podolski is now set for a prolonged spell of rehabilitation, which will further stretch Wenger's already depleted squad.
Speaking to the club's official website, www.arsenal.com, Wenger said: "It is bad news about Lukas Podolski as he will be out for eight to 10 weeks.
"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 [over Fenerbahce]. We have to be patient now."
Midfielder Aaron Ramsey scored twice against the Turks, but finished the match on the bench with a groin problem after Arsenal had used all three substitutes, while Jack Wilshere's fragile ankle was a cause for concern after he was the victim of some meaty tackles.
However, Wenger suggested both men should be fine for Sunday's north London derby against Tottenham.
He said: "The news is quite positive. They just had bumps and bruises. They will be involved."
Midfielder Mikel Arteta and captain Thomas Vermaelen continue their recovery.
Wenger added: "Arteta is doing okay, but there are few weeks to go with him. Thomas Vermaelen could be back after the international break."
Arsenal are expected to move ahead of the September 2 transfer deadline, with Real Madrid winger Angel di Maria one of several targets.
The Gunners are also said to be close to a deal to re-sign former midfielder Mathieu Flamini, 29, who has been training with them since his release by AC Milan.