The Germany international suffered the injury during Tuesday night’s 2-0 win over Fenerbahce in the Champions League play-off at Emirates Stadium.
Podolski was substituted after 49 minutes and it was initially feared he would be sidelined for up to three weeks.
However, at his weekly press briefing ahead of Sunday’s North London derby, Arsene Wenger said the prognosis was much worse.
“It is bad news about Lukas Podolski as he will be out for eight to 10 weeks,” Wenger said.
“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.”
Podolski started the season on the bench but had scored twice in a win over Fulham at the weekend to suggest he was ready to reclaim a regular place in the first team.
Aaron Ramsey, who scored twice against Fener, and Jack Wilshere are both fit for Sunday's home game with rivals Tottenham but Podolski's injury still means Arsenal are desperately short of players just a handful of games into the season.
The Gunners are believed to have offered around £26 million for Real Madrid winger Angel Di Maria and Mathieu Flamini is poised to rejoin the club on a free transfer, but the need for new signings is ever more acute in light of the Podolski news.