The 32-year-old has just entered his fourth season with the Merseyside giants, and the versatile Englishman has become an increasingly influential figure in Jurgen Klopp's side.
However, according to the Daily Mail, no talks have taken place about improved terms for a player who has less than two years remaining on his existing deal.
Milner will be 34 years of age at the end of the 2019-2020 campaign, but Liverpool are likely to want to extend the former England international's stay at the club after emerging as one of the club's key men.
Klopp has handed Milner a starting role in all six of Liverpool's fixtures in the Premier League and Champions League since the start of the new campaign.
Milner impressed on Tuesday night as Liverpool registered a 3-2 victory over Paris Saint-Germain at Anfield.