The Reds are believed to have made the England international – who will travel to the World Cup – a top priority, and want a deal tied up quickly.
Negotiations between the two clubs have been bubbling away over the course of the past few weeks, but the Saints have thus far rejected a £20m offer from Brendan Rodgers’ outfit.
Southampton will have to pay Bournemouth 25% of any fee due to a clause in the deal that took him across the South Coast as a youngster, and want Liverpool to foot that bill by offering more cash.
The Merseysiders have relented in their valuation as a result, and a final bid of £25m has been lodged.
It is understood that if they are unsuccessful Rodgers will look elsewhere, with Bayern Munich’s Xherdan Shaqiri an alternative.
Lallana netted nine goals in 38 Premier League outings last term from an attacking midfield role, and was, arguably, Southampton’s most consistent performer in a good season.
The 26-year-old may follow Rickie Lambert to Anfield, with the 32-year-old forward set for a medical today ahead of completing a shock £4m move to his boyhood club.