Milner, 37, has been linked with moves to Brighton and hometown club Leeds after spending eight years at Anfield.
He made his Premier League debut for Leeds aged 16 in 2002 and also had spells at Newcastle, Aston Villa and Manchester City before joining Liverpool in 2015.
He has made 617 Premier League appearances in total, behind only Gareth Barry (652) and Ryan Giggs (632) on the all-time list.
The midfielder overtook Frank Lampard when making his 610th top-flight appearance early last month in a 0-0 draw at Chelsea.
Liverpool signed Milner on a free transfer from City, with whom he won two Premier League titles, an FA Cup and the League Cup during his five years at at the Etihad Stadium.
In his eight seasons with Liverpool, he won the Champions League, Premier League, FA Cup, League Cup, UEFA Super Cup and FIFA Club World Cup and made the last of his 61 appearances for England in 2016.
Firmino has scored 109 goals in all competitions for the Reds since joining from Hoffenheim, including 11 in 33 games this season.