The Hammers are looking at left-backs, after discovering Aaron Cresswell could miss the first four months of the new Premier League season. The star defender has suffered a serious knee injury during pre-season.
Enrique was frozen out of the Liverpool squad during the last two seasons, managing just a combined 12 Premier League appearances across the last three campaigns, without featuring in any league game last year. He left the club this summer after his contract expired.
The IB Times claim Tony Pulis is keen to sign him for West Brom, after previously trying to sign the Spaniard last summer. The Baggies are also looking for a new left-back, after having to use both Chris Brunt and Jonny Evans in the position last year as stop-gaps.
Crystal Palace have been linked with Enrique too, with Alan Pardew previously working with the 30-year-old during their time at Newcastle United together.
La Liga teams Malaga and Espanyol are also interested.