The club have already held talks with Liverpool and Lambert concerning a move to the Hawthorns, and it is stated that progression has been ‘positive’, although the transfer is on hold until next week.
Lambert, 33, is currently away with the Reds on their pre-season tour of Asia, and they head back to England at the weekend, following which a deal is expected to be concluded.
The former Southampton forward only signed for the Reds last summer, but due to the arrivals of Christian Benteke, Danny Ings and Divock Origi at Anfield, his place in the team is under threat.
First-team chances would be almost impossible to come by for the striker, and with just 12 months left on his contract, Liverpool want to sell him.
The transfer fee is stated to be roughly £3 million, as per the report, and with all the framework in place for a deal to be completed, Lambert should be finalising his move to the Midlands club at the beginning of next week.
This means the Baggies are not actively pursuing a deal for Borussia Dortmund striker Adrian Ramos, although the report indicates that they will continue to track him in case the Bundesliga club lower the asking price on his head.