The Germany international is reported to have travelled to the Maxx Royal Kemer hotel in the Mediterranean coastal city of Antalya.
Podolski is expected to hold direct talks with Galatasaray while in Turkey and join the Istanbul giants.
Galatasaray have made Podolski an offer worth €3.5 million per-season according to the Sabah.
The World Cup winner also recently released a statement confirming that he would consider a move to Turkey as well.
The Germany international winner 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 unlikely to stay at Arsenal after falling out of favour with Gunners boss Arsene Wenger but still has a year left on his contract.
The experienced striker won the World Cup with Germany last season and wants regular first team football to keep his place in the national team side.