Atalanta was also reported to be interested in the Japan international, who has struggled for regular first-team action under Jurgen Klopp, while Wolves and Fulham were also reported to be interested earlier this month.
However, Sportsmail understands that Monaco, who has recently lost Cesc Fabregas, is now in the driving seat after agreeing to an initial £12.9m fee plus a further £2.6m in add-ons with the Reds.
Minamino was a vital figure in the Reds’ FA Cup and League Cup triumphs last season, scoring seven goals in domestic cup tournaments.
In 2021-22, though, he only made 11 Premier League appearances for Jurgen Klopp’s side, scoring three goals in the process.
Minamino joined Liverpool for £7.25 million from Red Bull Salzburg ahead of the January 2020 transfer window.