Tony Pulis is eager to bring in a central midfield player this month and had earmarked Morgan Schneiderlin as his leading target.
But Schneiderlin opted to join Everton from Manchester United, forcing Pulis to look elsewhere.
Now the Baggies boss has set his sights on Livermore.
The 27-year-old joined Hull from Tottenham in 2014 for a fee of around £8million after a successful loan spell and has made 144 appearances for the club, scoring nine goals,
Livermore had started every Premier League this season prior to Saturday's win over Bournemouth, when he was surprisingly left on the bench by new boss Marco Silva.