Hamzaoğlu revealed that the club will target a striker over the summer transfer window.
“Podolski and Gignac are both on our shortlist we have not entered direct talks,” Hamzaoğlu told NTV Spor.
Podolski recently stated that he would consider a move from Turkey and Galatasaray have been closely linked with the Germany international.
The Lions made an offer in January but Podolski joined Inter on loan instead.
Podolski told NTV Sport: “If an offer comes from Turkey, I will sit down and talk about it with my family and girlfriend.
“If I find the right club, the right coach, the right environment, I will evaluate it.
“If these conditions are met, I think everything is possible.
“The next transfer could be my last, so I have to take a well-thought step.”
Podolski is expected to leave the Gunners in the summer and Fenerbahçe have also been linked with the World Cup winner.
Hamzaoğlu also confirmed that Goran Pandev will be leaving over the summer transfer window.
“We have parted ways with Pandev and we are in talks with Dzemaili,” he added.
Pandev joined Galatasaray from Napoli last season but failed to settle in and struggled for playing time.