The former Stoke and Crystal Palace chief – who was named the Premier League’s star manager last term – recently took over the Baggies following the exit of Alan Irvine, and is understood to have funds in place to bring his own players to the Midlands.
Although record signing Ideye Brown and Saido Berahino are available to him up front, Pulis wants more bodies to boost his attack, with Deeney an option.
West Brom have reportedly lodged an enquiry for the Watford ace, who has bagged 69 goals for the Hornets in just under five years.
The 26-year-old – who is considered to be one of the Championship’s leading marksmen – is also a target for Swansea, who are set to lose Wilfried Bony to Manchester City in a £30m deal.
Vicarage Road chiefs are reluctant to sell the forward, but the lure of top-level football may sway the player.