Deeney has fallen down the pecking order at Vicarage Road this season, making just eight Premier League starts under Marco Silva before the Portuguese manager was sacked earlier this month.
New boss Javi Gracia also omitted the 29-year-old from his squad entirely for his first match at the helm on Saturday as the Hornets fell to a 1-0 FA Cup defeat at the hands of Southampton.
Watford are now thought to be open to the prospect of their skipper leaving the club this month and are willing to lower their initial £25m asking price, with Deeney - who will turn 30 in June - one of the club's highest earners on around £70,000 a week.
The Daily Mail reports that West Brom are planning to test their resolve with a bid of £18m as Alan Pardew looks to bolster his attacking options having seen Hal Robson-Kanu pick up an injury during Saturday's win over Liverpool.
The Baggies would likely need to sell before they could afford the striker, but the likes of Jonny Evans and Jake Livermore have both been linked with moves away from The Hawthorns before Wednesday's deadline.