The 19-year-old is expected to leave Russell Martin's side after just one season following the Saints' relegation to the Championship for the 2023-24 campaign.
Lavia failed to make the grade at Manchester City and made the move to Southampton for £10.6m in 2022, contributing one goal and one assist from 34 games in his debut season.
The Belgium international still has four years left to run on his deal with Southampton, but a summer exit is seen as inevitable, assuming that an interested party meets his £50m price tag.
Over the weekend, it was suggested that Liverpool were prepared to up their offer to £55m in an attempt to land the Belgian starlet, but according to The Guardian, the Reds have instead lodged a £60m bid.
Southampton accepted the offer from their Merseyside counterparts, who had previously refused to break the £45m barrier, but the midfielder's proposed move to Anfield is still far from a formality.
Liverpool have accelerated their interest in Lavia after conceding defeat in the race for Brighton & Hove Albion's Moises Caicedo, who is set to join Chelsea in a British-record £115m deal.
The Reds had agreed a deal with Brighton to sign the Ecuadorian for £110m, but Chelsea raised their nine-figure offer after learning that Caicedo had his heart set on a move to Stamford Bridge.
While the Blues are breaking the bank to sign the former Independiente del Valle man, Mauricio Pochettino's side are also hoping to hijack Liverpool's move for Lavia, who was apparently the subject of a £55m offer from Chelsea last week.
In another potential blow to Jurgen Klopp, the report adds that Lavia is leaning towards a move to Chelsea, but the German will fight tooth and nail to bring him to Anfield as they desperately search for a new number six.
Fabinho ended his glittering five-year spell at Anfield to join Al-Ittihad in a £40m deal earlier this summer, while 18-year-old protege Stefan Bajcetic is still recovering from an adductor injury.
All of Jordan Henderson, Leighton Clarkson, Naby Keita, Alex Oxlade-Chamberlain and James Milner have also moved onto pastures new, while Thiago Alcantara is struggling with a hip concern.
As a result, Klopp was forced to field an attack-minded midfield of Alexis Mac Allister, Dominik Szoboszlai and Cody Gakpo against Chelsea on Sunday, where Luis Diaz's strike was cancelled out by Axel Disasi.
The Reds introduced Curtis Jones in the second half for more midfield stability, but Klopp's side had just 34.6% of the ball at Stamford Bridge, their lowest possession percentage versus the Blues since Premier League records began in 2003-04.
Klopp's side are hoping to welcome Lavia to the club in time for their second Premier League fixture of the season, which comes at home to Bournemouth next Saturday.