The 19-year-old moved to St Mary's from Manchester City last summer for £14m, impressing during his first season in senior football despite Southampton's relegation from the Premier League.
Lavia made 29 appearances in the English top flight for the Saints - 26 of which came from the start - and he managed to net his maiden goal in first-team football against Chelsea in August 2022.
Commendable performances on the South Coast led to the youngster being selected for the Belgium national team during their March internationals, and the midfielder made his debut for his nation during a friendly win over Germany.
According to Sky Sports News, Liverpool's pursuit for Lavia looks set to drag on late into the window, with Southampton dismissing a third offer for the player in the region of £45m.
It has been widely reported that the second-tier Saints are hoping to pocket £50m from the sale of the 19-year-old - a fee which would make Lavia the Championship's most-expensive outgoing of all time.
The report also claims that Chelsea are monitoring Lavia's situation as a St Mary's departure appears likely, with Mauricio Pochettino looking to bolster his engine-room options following the exits of Mason Mount and N'Golo Kante.
Manchester City also have a horse in the race, with the treble-winning Citizens set to receive 20% of the transfer fee collected by Southampton for the youngster this summer.
Jurgen Klopp is undergoing a much-needed midfield rebuild at Anfield, recruiting Alexis Mac Allister from Brighton & Hove Albion for £35m and signing Dominik Szoboszlai from Bundesliga outfit RB Leipzig for £60m.
It is believed that the Reds are in the market for further reinforcements in that area of the pitch, especially considering the recent departures of Fabinho and former captain Jordan Henderson to the Saudi Pro League.
It was been reported elsewhere that Liverpool have submitted a £21.5m bid for Celta Vigo's Gabri Veiga, with Chelsea and Ligue 1 champions Paris Saint-Germain also keen on the Spain Under-21 international.
Lavia was part of the Southampton squad which travelled to Hillsborough for their second-tier opener against Sheffield Wednesday on Friday night, although the Belgian was an unused substitute against the Owls amid intense transfer speculation.
West Ham United-linked James Ward-Prowse started in the Saints' engine room alongside William Smallbone and Carlos Alcaraz, however it remains to be seen whether the England international will be at the club by the end of the window.
An attack-minded midfield three of Mac Allister, Szoboszlai and Cody Gakpo started Liverpool's final pre-season friendly against German side Darmstadt on Monday night, although Klopp is likely to be more conservative in his selections when they travel to Stamford Bridge in their Premier League opener this weekend.