The German forward joined the Istanbul based side for £1.75 million last month following an unsuccessful loan spell with Inter Milan.
Mertesacker went on to wish the 30-year-old good luck at his new club.
“It is a shame to lose Lukas as a team-mate,” Mertesackler told transfermarkt.de (via Goal).
“His left foot is a weapon which I did not see with many players. On his day, he can still make the difference. I wish him good luck in Turkey.”
Podolski won the World Cup with Germany in 2014 and wants regular first team football to keep his place in the national team side.
Galatasaray won the Super Lig for the 20th time last season and earned the right to wear the fourth star next season.
The Lions booked a place in the Champions League group stages for the fourth consecutive season after winning the championship.