The Merseysiders submitted a €25m bid for the England international on Monday only hours after the end of the Premier League season.
The Saints accept that Lallana is ready to push for a move to Liverpool, who will compete in next season's Champions League after finishing second, but value their skipper far higher than the offer on the table.
They are expected to reject the Reds' bid and hold out for a transfer fee closer to their €37m valuation of the player.
|11/10||Adam Lallana is 11/10 with PaddyPower to start for England against Italy at the World Cup|
The attacking midfielder, who turned 26 last Saturday, has attracted interest from Chelsea, Manchester United and Tottenham but has been most strongly coveted by Liverpool, who have made him one of their main summer targets.
Reds manager Brendan Rodgers is a huge admirer of the Saints star's technical skills and ability to play in a number of attacking positions behind the strikers.
Lallana is contracted to the Hampshire club until 2018 after signing a new deal last year but those close to him believe that the next window could be an ideal time to move to one of England's biggest clubs.
Southampton have publicly insisted that they are under no pressure to sell players but they are believed to be prepared to cash in on some of their prize assets, who include not only Lallana but Luke Shaw, Jay Rodriguez and Morgan Schneiderlin. Shaw is close to tying up a €30m-plus move to Manchester United, where he could become the world's highest-paid teenager, while Schneiderlin is a long-time target for Arsenal.
New Southampton director Hans Hofstetter admitted in March that the restructured board had "inherited a difficult situation financially" as it emerged in the club's latest set of annual accounts that there was an outstanding transfer debt of €33m.
Lallana, who joined the Saints as a 12-year-old and graduated from their famed academy, has had an outstanding season for Mauricio Pochettino's team, in which he started 37 of their 38 Premier League matches and scored nine goals.
Pochettino spoke in glowing terms of his club captain in the wake of Southampton's 4-0 win over Newcastle in April, describing him as a "special and unique player" who “has the skills of an [Andres] Iniesta, Xavi or [Cesc] Fabregas".