According to La Nuova Sardegna (h/t Football Italia), the Reds have sent scouts to watch the 21-year-old, who made his Italy debut earlier this week.
Corriere dello Sport and Tuttosport (h/t Football Italia) reported the Milan duo's interest, with Cagliari having priced Barella at €50 million (£44 million).
Despite his youth, Barella has been a stalwart for Cagliari in Serie A for the past three seasons and already boasts 73 appearances in the Italian top flight.
He most frequently operates as a box-to-box central midfielder, but he can also play in a more advanced or defensive role.
A Cagliari native, Barella came through the Rossoblu's youth ranks and, aside from half a season on loan in Serie B with Como in 2016, he has always played for the club.
Barella has turned out for Italy at every youth level since under-15s. He impressed in his senior debut on Monday in a 1-1 draw with Ukraine, starting alongside Jorginho and Marco Verratti on the left side of a central three.
Jurgen Klopp's Liverpool are not necessarily short on central midfielders after a summer transfer window in which they signed Fabinho and Naby Keita.
But they still do not boast the required depth to compete consistently for trophies on multiple fronts, and Barella would be an exciting young signing who could thrive under the German manager.