The England international is expected to be allowed to leave Anfield, with the Merseyside outfit activating Aston Villa striker Christian Benteke's £32.5m buy-out clause, as reported by the Mirror.
However, the Baggies will have to be patient if they wish to land the 33-year-old, with Liverpool and Lambert expected to make a decision after the club's pre-season tour.
Ongoing takeover talks could also delay a potential deal, with the club unable to compete with the increasing size of transfer fees and contracts demands.
The same report suggests that West Brom are also targeting Carlton Cole, who they tried to sign in January when he a West Ham player - he is now a free agent.
What do the stats say?
Despite making 25 appearances for Liverpool last season, Lambert was only afforded seven starts by manager Brendan Rodgers. This lack of game time saw him score on just two occasions - converting 13.3% of his total chances - while fashioning 11 key passes in assistance of his team-mates. He also completed just 68% of his passes, while finding success in only 43% of his aerial duels.
It is the statistics from the season before - during his time at Southampton - that will have attracted the interest of West Brom.
Indeed, during the 2013/14 campaign, Lambert made 37 appearances for the Saints, netting an impressive 13 goals - finishing the season as the club's second highest goalscorer, behind Jay Rodriguez. He also fashioned the third-most chances (53), while recording a 70% pass completion rate.
Whether Lambert can replicate the level of performance that he gave during his time at Southampton remains to be seen. If he can, he would become a very valuable player at the Hawthorns. An experienced head, who knows how to put the ball in the back of the net, the England international could prove a very useful signing for Tony Pulis' side.