Liverpool are understood to have increased their offer for Saints captain Adam Lallana to £25m.
The Reds had lodged an initial £20m bid shortly after the end of the season, and have now upped that as they try to strike a deal before the World Cup.
Liverpool have already agreed a fee with Saints for Rickie Lambert, who is set to have a medical tomorrow ahead of his proposed transfer to Anfield.
They had issued Saints with an ultimatum over that deal, saying it had to be sorted before England left for their pre-World Cup training camp in Miami.
Feeling that it could do more harm than good to deny Lambert a dream move to the club he supports, they reluctantly agreed to allow it to proceed.
That was despite chairman Ralph Krueger stating during the week that all transfer dealings were on hold until a replacement was found for Mauricio Pochettino, who walked out to join Tottenham last Tuesday.
Now, perhaps unsurprisingly, Liverpool are attempting to use the same tactic with Lallana, who starred in England's 3-0 World Cup warm up win over Peru at Wembley last night.
The Reds are understood to be saying this is a final offer and that, if a deal is not completed quickly, they will move on to other targets.