New Baggies boss Tony Pulis is desperate to bolster his front line in a bid to keep them in the Premier League, and has set his sights on Lambert.
The 32-year-old has barely featured for the Reds since sealing his dream move from Southampton in the summer.
Lambert has started just five league games, scoring one goal, coming off the bench in 11 games.
And Brendan Rodgers is understood be looking at other striking options this month.
