Tony Pulis sees the Hull City midfielder as the ideal man to strengthen West Brom's midfield after seeing an attempt to sign Morgan Schneiderlin from Manchester United fail.
The Baggies have launched a £10m bid for the former Spurs man, which Hull are currently pondering.
Livermore has been a key figure for Hull since joining from Tottenham in 2014, playing 144 times for the club and starting in FA Cup and play-off finals.
The 27-year-old also has one cap for England, and has been utilised as a centre-half on a number of occasions this season, meaning that he would add valuable versatility to the West Brom ranks.