The Ligue 1 side were one of a number of clubs interested in the 27-year-old, who joined the Reds from Red Bull Salzburg for £7.25m in 2020.
The deal for Minamino is worth a fixed 15m euros (£12.9m) and 3m euros (£2.6m) in add-ons.
He scored 14 times in 55 outings for the Reds and was their top scorer in domestic cup competitions last season.
Minamino's seven goals were crucial as Jurgen Klopp's side won the FA Cup and League Cup.
However, he made just 11 appearances, including one start, in the Premier League as Liverpool finished a point behind Manchester City.
Minamino also joined Southampton on loan in February 2021 and made 10 appearances, before returning to Liverpool for the 2021-22 campaign.
Senegal forward Sadio Mane is also close to leaving Liverpool after the Merseysiders agreed a £35.1m fee, which includes a fixed £27.4m for the 30-year-old, with German club Bayern Munich.
The expected sales follow the arrival of Uruguay striker Darwin Nunez from Benfica for an initial £64m.
Liverpool have also brought in forward Fabio Carvalho from Fulham for £5m and 18-year-old full-back Calvin Ramsay from Aberdeen for £4.2m, with both deals potentially increasing with add-ons.