The Reds launched an improved offer for the midfielder last night, in an attempt to force a move through quickly.
Liverpool had initially offered £20m for the 26-year-old, but were unable to convince Saints to sell for that amount.
They are understood to have said that their latest approach is a final one, and have threatened to move onto other targets if the transfer is not concluded before the World Cup.
That is the same tactic they used to help push through a deal for Lallana's club and international teammate, Rickie Lambert, who is expected to complete his switch to Anfield on Monday.
However, Saints feel they are in a stronger position when it comes to Lallana, and also Luke Shaw, with Manchester United making their own final push to sign the left-back before England depart for Brazil.
United have already offered £27m for the 18-year-old and are keen to wrap up a deal before anyone else can swoop in and hijack it, approaching Saints again yesterday to try and move things forward.
The club want to put transfer dealings on hold until a new manager is in place and the World Cup is over.
That stance was undermined somewhat by the agreement to sell Lambert, and it would be completely blown apart if they let either Lallana or Shaw go in the coming days.
It is increasingly unlikely that either player will remain at the club this summer, but Saints feel under less pressure to deal with those matters quickly than they did with Lambert.
Even if they do feel a sale is inevitable, the club know that interest will not disappear and their values may even rise in Brazil.
Roy Hodgson's squad fly to Miami tomorrow for a pre-World Cup training camp, before jetting off to South America next Sunday.
Hodgson has said he will allow clubs to send out staff to Florida to conduct medicals in the players' free time if absolutely necessary, but nothing will be allowed once England head to Brazil.
That leaves just a week for any transfers to be completed, although even once the squad depart tomorrow it will become increasingly difficult.