The Reds are believed to be in discussions with the player for a £4m plus add-ons deal.
Bookmakers have even suspended betting on the move, indicating that Lambert could well be a Liverpool player for the 2014/15 season.
The 32-year-old England international – who is set to represent the Three Lions at the World Cup – has been one of the Premier League’s more impressive strikers in recent seasons, earning much praise for his adaptation to the top tier after spending much of his career in the lower divisions.
Lambert is known to be a massive fan of the Anfield outfit, and still loves the club despite being released by them as a 16-year-old.
Liverpool are also chasing Adam Lallana and Dejan Lovren, who both enjoyed successful seasons with the Saints.
Should the move go through, Lambert is likely to play a back-up role to Luis Suarez and Daniel Sturridge and offer the Merseysiders a different attacking option.