Lambert faces stiff competition for a starting place at Anfield next season following the summer additions of Christian Benteke, Danny Ings and Roberto Firminho.
Baggies boss Tony Pulis completed the signing of James Chester from Hull City on Wednesday and is hopeful of following up that business with the capture of the former Southampton striker.
“Rickie is available,” the Albion boss told the Birmingham Mail. “Liverpool have made that known. We’d be interested.
“It’s very important that we get players who will score goals. Rickie’s got a great goal-scoring record.
“From what I can gather from people at Liverpool, his fitness levels are very, very good, and he is desperate to play football.
“He didn’t play last year so it’s like bringing a brand new car out of the garage if we did sign him. Just putting a bit of petrol in it and letting it run.”
Lambert had just two Premier League goals to his name from 25 games for Liverpool last season, 18 of those off the substitutes’ bench.
But the 33-year-old was more prolific at previous club Southampton where he totalled 106 from 207 league appearances from August 2009 until June 2014.