Mail Sport reported earlier this week that the club were preparing a revised bid shy of £50m, Lavia’s price tag, after their opening offer of £34m plus £4m was rejected.
Jurgen Klopp is believed to be keen to bring the talented teenager to Anfield this summer and an improved offer may tempt Saints into a reluctant sale.
Lavia, who moved to St Mary’s from Manchester City last year, is in demand ahead of the new season after impressing during his debut campaign on the south coast.
As well as Liverpool, Chelsea are also understood to retain an interest.
The 19-year-old featured in 29 Premier League games last season and proved to be a shining light in a year that saw the club condemned to the Championship.
He is likely to leave over the coming weeks, however, and Liverpool, in their bid to further strengthen their midfield, are hopeful of securing his signature.
Klopp has already added Alexis Mac Allister and Dominik Szoboszlai to his squad but after losing several midfielders remains in the market for more additions.
Captain Jordan Henderson departed for Saudi Arabian side Al Ettifaq, managed by Steven Gerrard, while Fabinho may also leave with Al-Ittihad closing in on a deal.