Reds manager Brendan Rodgers is said to have made an offer of £4 million ($6.6 million, 4.9 million euros) plus add-ons for Lambert and the BBC says talks are continuing to seal the deal after the initial bid was rejected.
The surprise move comes as Lambert prepares for England’s World Cup warm-up against Peru at Wembley on Friday.
Once his England duties are completed, the Daily Mail reports Lambert could undergo a medical on Saturday with a view to sealing the transfer before he jets off with the England squad for their pre-World Cup training camp in Miami.
It would be a dream move for Liverpool-born Lambert, who supported the club as a child and was a member of the Reds’ youth academy for five years before being released as a 15-year-old
The 32-year-old, who worked in a beetroot factory before his professional career finally got going, spent much of his career in the lower leagues of English football with the likes of Rochdale, Stockport and Bristol Rovers before finally hitting the big time with Southampton following their promotion to the Premier League.
He scored 14 goals in 39 games this season and has 117 in 235 appearances since joining Southampton in 2009.
Lambert’s potent combination of height, strength and clinical finishing would make him an intriguing addition to Rodgers’ squad, which noticeably lacked quality cover for first-choice strikers Luis Suarez and Daniel Sturridge this season.
While Suarez and Sturridge nearly fired Liverpool to the Premier League title, the Reds often looked toothless when their star strikers were held in check and Lambert would be an ideal remedy to that problem.
“I have been a big admirer of Rickie,” Rodgers said earlier this season. “He’s never got the recognition for what a really good footballer he is.
“He is probably seen as a traditional big number nine, a typical British striker, but he’s one of the most accomplished footballers I’ve seen.
“Look at his touch and the different types of goals he scores; he is a terrific player.”
If Lambert’s switch goes through, he could soon be joined at Anfield by Saints team-mate Adam Lallana, with Liverpool currently trying to negotiate a £30 million deal for the England midfielder.