Lavia has been heavily linked with a move to the Reds all summer, and they have also had a handful of offers turned down by the Saints.
It looked as though the Merseysiders would have more leeway to pursue the Belgian midfielder after missing out on Brighton & Hove Albion midfielder Moises Caicedo in a dramatic transfer saga.
The Reds reached an agreement with the Seagulls, outbidding Chelsea with a British record fee of £110 million.
However, the Ecuadorian midfielder has made it clear that he wants to move to Stamford Bridge, causing the Reds to pull the plug on the deal.
The west Londoners are not done yet, as they also want to add Lavia to their ranks in what could be back-to-back blows for Liverpool.
According to journalist Sacha Tavolieri, the Blues have tabled an improved offer of £52m plus £3m in add-ons to the Saints for Lavia.
Chelsea are looking to wrap up the deal and schedule the medical tests soon to thwart any last-minute hijack attempts from Liverpool.
Recruitment director Joe Shields has played a decisive role in the deal as his healthy ties with his former club gave the Blues an upper hand.
Chelsea appeared to be lingering over Caicedo’s deal, but they are now close to completing two blockbuster transfers.
Midfield reinforcements have been a pressing priority for Pochettino’s side following the departures of Mateo Kovacic, Ruben Loftus-Cheek and N’Golo Kante.
Chelsea will kickstart their 2023/24 Premier League campaign with a heavyweight clash against Liverpool on Sunday.
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