The Hammers chief has watched his side register a measly 18 goals this term, which has contributed heavily to their current spot in the Premier League’s bottom three.
Big money summer signing Andy Carroll has been unable to make any impact since his switch from Liverpool, while Modibo Maiga and emergency signing Carlton Cole have failed to ease the blow of losing the burly Geordie.
‘Big Sam’ has made a new hit-man his main priority for the winter window, and has outlined Lambert as his top target.
The Irons boss has already made an opening enquiry for the 31-year-old, which has been met with an £8m price tag from Southampton.
The Merseyside-born forward has been a key member of the Saints’ rise from League One to the top tier, but has recently found his starting role challenged by Pablo Osvaldo.
As a result he may be keen to make a January move to London, with West Ham likely to present him with regular playing time ahead of the 2014 World Cup – for which he may make Roy Hodgson’s England squad.