The former Brazil international moved to Merseyside in summer 2007, going on to make 313 appearances across all competitions and lift the League Cup in 2012.
But the tough-tackling midfielder looks set to leave Anfield before the close of the summer transfer window, with his contract due to expire at the end of next season.
Recent reports have linked the 29-year-old with a move to Newcastle United, which would see him link up with former Reds boss Rafa Benitez.
But according to Fanatik, Leiva has now agreed a three-year contract with Galatasaray that will see him take home around £2.1million per season.
The Super Lig outfit have already tabled a £2.5million bid, although it remains unconfirmed whether the Reds have accepted it.
Leiva is set to miss the start of next season with a hamstring injury, according to The Liverpool Echo.