The 29-year-old will travel south later on Tuesday to undergo a medical and discuss personal terms ahead of a potential move to The Hawthorns, according to Sky Sports.
Enrique joined Liverpool – then under the management of Kenny Dalglish – from Newcastle in 2011, but a succession of injuries has curtailed his spell at Anfield.
The arrivals of Alberto Moreno last summer and Joe Gomez in June has seen Enrique fall down the pecking order under manager Brendan Rodgers and he is yet to feature for Liverpool this season.
West Brom manager Tony Pulis has already strengthened his defensive options during the summer transfer window with the signings of James Chester and Jonny Evans from Hull City and Manchester United respectively.
If Enrique successfully passes his medical and agrees personal terms, he will become Pulis’ eighth acquisition of the summer, following the other arrivals of James McClean, Rickie Lambert, Serge Gnabry, Salomon Rondon and Anders Lindegaard.
Pulis is also said to be chasing Walters, who has handed in a transfer request at Stoke, and Wilson, who Manchester United may allow to leave on loan.
Albion are also in talks over a £5million move for Napoli midfielder David Lopez, according to the Daily Mail, but face competition from Leicester.
Aside from the three new names, QPR duo Charlie Austin and Matt Phillips are long-term targets of Pulis’.