The Reds forward looks set to move on after just one season at Anfield.
Sky suggest Lambert will sign at the Hawthorns later on Friday as long as there are no problems that arise in the medical.
West Brom boss Tony Pulis had earlier in the week praised the prospect of signing Lambert, despite his struggles last season.
"It's very important that we get players who will score goals," the Welshman told the Birmingham Mail. "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."
The right move for Lambert?
Including Lambert, Liverpool have a surplus of forwards on their books, with several players struggling to make an impression during their time at Anfield.
Lambert is one of those players that struggled, as difficult as it would be for the native Liverpudlian to admit of his local team.
The 33-year-old scored just three goals all season after arriving from Southampton, and was often failing to earn minutes on the pitch by the end of the campaign despite the club continuing to have issues finding the net.
However, he remains a viable option for any Premier League clubs who are not restricted by having a short memory, after a prolific spell at the Saints that led all the way from League One to the top flight, as well as an impressive start to his England career.
Tony Pulis will certainly be hoping that he can rediscover that form that he had at St Mary's as soon as possible.