Anelka endured an unhappy spell with Shanghai Shenhua before joining Juventus on loan last term.
And he is now thought to be keen on a move back to the Premier League, with West Ham having also been linked with the former Arsenal and Chelsea man earlier this summer.
The Hammers are now out to 7/2 to sign the 34-year-old on the back of punters' interest in West Brom, with Crystal Palace (16/1), Fulham (16/1) and Southampton (33/1) thought only to have an outside chance.
Sky Bet's Chris Spicer said: "Whilst the number of stakes hasn't been particularly high, the sheer size of them has taken us somewhat by surprise.
"Our customers have lost interest in West Ham since they wrapped up a deal for Andy Carroll, so it appears the path is clear for the Baggies."
Should West Brom boss Steve Clarke seal the move, the Baggies would become Anelka's sixth Premier League club after Arsenal, Liverpool, Manchester City, Bolton and Chelsea.