Tony Pulis is keen on the striker, 33, and has sanctioned a move which will see the Baggies move to sign Lambert this summer.
According to the report, Lambert has been told he can leave Anfield this summer - following the arrivals of Danny Ings, Divock Origi and the soon-to-be arrival of Christian Benteke - and although he rejected a move to Aston Villa in January, a transfer this summer seems likely.
There are no indications as to whether Liverpool will be willing to accept a £3 million fee, but the Baggies believe it is enough to sign a player who will have limited first-team chances next season.
The Reds may also accept the fee, as it is a large price for a 33-year-old player who has never excelled at Anfield, and whose only redeeming factor for the club is his homegrown status.
Pulis has become frustrated with the lack of transfer business seen at the Hawthorns this summer, and has held talks with chairman Jeremy Peace to speed up any incoming arrivals.
Moves for QPR’s Matt Phillips and Nottingham Forest’s Michail Antonio look to be stalled, and it is believed that Pulis wants to have at least one more signing made before this weekend.
That man looks to be Lambert, and if Liverpool are as willing to sell the striker as believed, then the £3 million fee should be enough to prise him away from Anfield.