Liverpool manager Brendan Rodgers has set out his transfer stall early in preparation for the new Premier League season.
The England international has travelled north after being an unused substitute for England in the 3-0 friendly win over Peru at Wembley on Friday night.
The 32-year-old was born in Kirkby, Liverpool, and was brought up as a Liverpool supporter, and even has a Liver Bird tattoo on his shoulder, according to James Pearce of the Liverpool Echo.
Former Liverpool striker Robbie Fowler has declared the capture of Lambert as a "magnificent signing," reports Vic Holly of The Sun (subscription required).
Fowler is quoted as saying:
Lambert would be a magnificent signing.
For £4 million, you’re getting a recognised striker who scores goals and who can hold balls up.
He’s what Liverpool need, he’s got a little bit of everything.
The Reds have also made a final offer of £25 million for Lambert's Southampton team-mate Adam Lallana, claims Ben Smith of BBC Sport.
The 26-year-old played in England's win over the Peruvians on Friday, and will be part of the squad which travels to Miami on Sunday.
Midfielder Lallana scored nine times in 38 Premier League appearances last season to help Southampton to an eighth-placed finish in the table.
However, Smith has also reported that the deal has reached a stalemate, with Lallana wanting the deal sorted before the World Cup begins, while the St Mary's Stadium club want to wait.
The 21-year-old has been cut from the Spain World Cup squad for Brazil, despite playing in the friendly against Bolivia on Friday night.
The Italian spent last season on loan with the Black Cats and helped Gus Poyet's side to a remarkable escape from relegation and to the Capital One Cup final against Manchester City.
But the signing of Lambert is likely to see Borini relegated to fourth-choice striker at Anfield.