The 33-year-old joined Liverpool from Manchester City on a free transfer in 2015 and has made 181 appearances for the Reds in four years at Anfield.
Jürgen Klopp considers Milner a valuable senior member of his squad and his ability to play in a number of positions is deemed crucial.
Milner, for his part, feels he has plenty left to offer on Merseyside and the Athletic report the midfielder has opened talks with the club over a new deal.
While an agreement has not yet been reached, more talks are set to follow and there’s an expectation on all sides that a resolution will be found.