Goal understands the Reds have launched a bid that could rise to £9 million as Brendan Rodgers looks to add to his attacking options ahead of the new campaign
The Reds have made an offer for the England international that could rise to £9 million as Brendan Rodgers looks to supplement his first-choice attacking line of Luis Suarez and Daniel Sturridge with proven Premier League quality.
Lambert's record of 13 goals in 37 games in England's top flight last season saw him finish above the likes of Robin van Persie and Alvaro Negredo in the scoring charts and just four behind Manchester United star Wayne Rooney.
Rodgers has been keen to bolster his attacking options to ease the goalscoring burden on Suarez and Sturridge, who scored 52 Premier League goals between them last season, with the Reds' other recognised striker Iago Aspas having struggled to make an impact at Anfield.
Lambert's strength and ability in the air will provide a different option for Rodgers, with the Southampton forward likely to be given plenty of first-team action as Liverpool look to cope with the demands of juggling Champions League football.
A move to the Merseyside club, whom he has supported since he was a boy, would represent the latest chapter in a remarkable 12 months for the Saints star.
The former Bristol Rovers striker, who has played in all four divisions in England, scored on his international debut against Scotland back in August and his performances for Southampton over the season have seen him earn a place in Roy Hodgson's World Cup squad.
And, with Lambert turning 33 in February, Southampton are unlikely to stand in the way of a dream move to Anfield, with a fee that could rise to £9m representing a significant investment on the figure in the region of £1m they paid to sign him back in 2009.