The Japan international, 24, is thought to have a release clause of £7.25m.
Minamino has scored nine goals and provided 11 assists in 22 appearances for Salzburg this season.
He played in both of his side's Champions League group games against the Reds this season, netting in the 4-3 defeat at Anfield in October.
Liverpool won 2-0 in Salzburg on 10 December to secure their place in the last 16 of the competition, where they will face Spanish side Atletico Madrid.
Despite playing against the Reds in this season's Champions League, Minamino would be eligible to play for the European champions in the knockout stages of the tournament following a rule change by Uefa last year.