The Japan attacking midfielder impressed against Liverpool in their two Champions League group games this season, scoring at Anfield.
The 24-year-old has a £7.25m release clause in his contract, which has alerted a number of Premier League, Bundesliga and Serie A clubs, so the Reds are accelerating their interest.
Minamino has been one of the stars of an exciting Salzburg team that has lit up the Champions League group stage this season, playing alongside the equally impressive forwards Hwang-Hee Chan and Erling Braut Haaland.
After going 3-0 down on Merseyside in October, head coach Jesse Marsch switched Minamino's position and he stole the show in a No.