According to the article, the Reds have agreed a fee with the south coast club, and personal terms with the 32-year-old.
The England international, who was on the bench for the 3-0 win over Peru at Wembley on Friday night, has arrived on Merseyside, claims Sky Sports News.
For Liverpool-born Lambert, the transfer represents a dream move having supported the Reds since childhood.
But is the Southampton and England forward the right choice in the transfer market for Brendan Rodgers?
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