Lallana enjoyed a superb season on the south coast, scoring nine goals during a campaign that saw him make his England debut against Chile last year.
The 26-year-old's form was rewarded with a spot in Roy Hodgson's 23-man World Cup squad in Brazil, and his displays have caught the eye of the Anfield club who are looking to bolster their ranks ahead of next season's Champions League campaign.
However, according to the Liverpool Echo, both clubs are still trying to thrash out a deal, with the five-time European Cup winners keen to secure the signing before the tournament kicks off in South America next month.
Lallana would provide Philippe Coutinho with fierce competition for the role of the side's advanced playmaker, and manager Brendan Rodgers is keen to boost his numbers with several quality additions as the club returns to Europe's premier competition for the first time in five years next term.
Liverpool enjoyed a memorable campaign last term, missing out on the Premier League title to eventual winners Manchester City by two points.
A rampant Reds side plundered 101 goals en route to amassing 84 points to jump from a seventh-place finish in 2013 to running City close a year later.