Former Napoli forward Mertens is currently a free agent after leaving the Serie A outfit at the end of last season, while Arsenal midfielder Torreira is in the final year of his contract with the Gunners.
"Official negotiations have started with the player regarding the transfer of professional football player Dries Mertens to our club," a Galatasaray statement issued via Twitter read.
"[Furthermore], negotiations have started with Arsenal regarding the transfer of Lucas Sebastian Torreira Di Pascua."
Mertens, a double Coppa Italia winner during his nine-year spell in Naples, was a member of the Belgium squad that came third at the Russia 2018 FIFA World Cup four years ago.
Despite being 35, he is yet to confirm his international retirement, and may hope to figure in the Red Devils' party for Qatar 2022 later this season.
Uruguay international Torreira meanwhile has spent the past two seasons away from Arsenal on loan, with Atletico Madrid, where he was a title winner in La Liga, and also Fiorentina in Serie A.