Anelka is a free agent and is thought to be keen on a move back to the Premier League, with West Ham having also been linked with the former Chelsea man earlier this summer.
The Hammers are now out to 7/2 to sign the 34-year-old, who spent the end of last season on loan at Juventus, 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.
Midlands newspaper the Express & Star claimed on Monday that Albion officials had spoken to the 34-year-old's representatives, with Baggies boss Steve Clarke having worked with Anelka at Stamford Bridge.
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 Clarke lure Anelka to The Hawthorns, the Baggies would become his sixth Premier League club after Arsenal, Liverpool, Manchester City, Bolton and Chelsea.