The England international has fallen down the pecking order at Manchester City and has been linked with an exit from the Etihad with Barca rumoured to be interested in the 26-year-old.
According to the Mirror, Sterling would reportedly like to play under Jurgen Klopp at Anfield rather than make the switch to Camp Nou.
Sterling previously made 129 appearances for the Reds, scoring 23 goals. He is contracted to City until June 2023, and is valued at £81m by Transfermarkt, despite a largely frustrating season so far.
He's started only three Premier League games this season and has 15 times in all competitions for City, scoring twice.
However, the report does add that City aren't open to the idea of strengthening a rival and will only sell Sterling to an overseas club.
Barcelona seem like the most likely destination for Sterling, but according to SPORT, the Catalan giants need to offload Frenkie de Jong, among others, to raise funds for the City man. And the Dutch midfielder has no intention of leaving Barca now that he is settled in Spain.