The 26-year-old former Wolfsburg forward joined the Turkish giants on loan from Napoli last season.
He scored 37 goals in 41 games in all competitions, alongside eight assists as his side won the national league and the Turkish Cup.
He has since netted two goals in three Super Lig matches this term.
Clubs across Europe and Saudi Arabia were said to be coveting the talented hitman, who ultimately chose to sign for Galatasaray on a permanent basis.
He told the club's website: "I've had big offers from some clubs there.
"I'd be lying if I said I wasn't interested; but I'm happy to be at Galatasaray.
“I already knew in my heart where I wanted to be. They valued me here… there is a really different understanding within the club.”
Arne Slot has said Liverpool's previous experience in Istanbul would help his players handle the "very impressive" atmosphere when they face Galatasaray in the UEFA Champions League last-16 on Tuesday night.
Ogenyi Onazi has criticised Manchester United's decision to drop Victor Osimhen from their transfer shortlist, insisting that fears over the Africa Cup of Nations were not a valid reason.
The UEFA Champions League round of 16 draw produced heavyweight clashes and high-stakes drama, with six Premier League clubs handed demanding tests on the road to Budapest.
Holders Paris Saint-Germain booked their place in the last 16 of the UEFA Champions League with a hard-fought victory over AS Monaco, in a dramatic contest that turned on a red card.
