The Saints striker has been in fine form for the newly promoted Premier League side this term, notching 11 goals, making him the division’s highest scoring Englishman.
According to reports, the Hammers made an approach for the 30-year-old last month, and could return with a summer bid should they fail to strike a deal for Andy Carroll.
The London club are believed to have agreed a permanent transfer clause with Liverpool for the on-loan front-man, but there are no guarantees that a deal will be completed.
His former club Newcastle are rumoured to be interested in bringing the man they sold for £35m in 2011 back to St James’ Park, and it’s likely that Carroll would prefer a return to the North East to an extended stay in the capital.
This has forced West Ham manager Sam Allardyce to search for alternatives, with Lambert believed to be high on his list.
He is confident that a deal can be reached with the Saints, who will be fully aware that their talismanic hit-man could leave for free at the end of next season when his contract expires.
One stumbling block could be Lambert’s willingness to stay with the South Coast club, with whom he enjoyed back-to-back promotions.