The Premier League champions' second offer for Wirtz includes add-on clauses that would make him the most expensive signing in the club's history.
Wirtz has also attracted interest from Bayern Munich and Manchester City, but it is believed he made it clear to Leverkusen that he wants to join Liverpool before next season.
Leverkusen sporting director Simon Rolfes on Monday said Liverpool had "concrete interest" in Wirtz.
Leverkusen CEO Fernando Carro had previously insisted he would not let Wirtz leave for less than £125 million.
The 22-year-old emerged as one of the most highly-rated young playmakers in Europe after helping Leverkusen win their first Bundesliga crown in 2023-24.
Wirtz was named Bundesliga player of the season for his role in that triumph.
He featured in 31 Bundesliga matches this season, scoring 10 goals as Leverkusen finished second behind Bayern in the title race.
Wirtz has scored 57 goals in 197 games since making his Leverkusen debut aged just 17 after signing from Cologne in 2020.