Albion boss Tony Pulis is expected to step up his bid to sign the Liverpool striker after returning from the club’s recent pre-season tour of the US.
Lambert has one year remaining on his current contract but has fallen down the pecking order since joining from Southampton last summer.
The 33-year-old, who has scored just three goals in 36 appearances for the Reds, is likely to see his chances of first-team football diminish further with the expected arrival of Christian Benteke from Aston Villa.
Lambert turned down a deadline day move to Aston Villa during the January transfer window to stay with the team which released him as a teenager.
Norwich City, AFC Bournemouth, Aston Villa and Middlesbrough have also been linked with the England international.