The former Blackburn forward has managed four goals in 13 Championship outings this season and The Sun claim he has caught Pulis' eye as he bids to bolster his Baggies attack in the winter transfer window.
Pulis is eager to add to his options up front amid doubts over Hal Robson Kanu, who has made all eight of his Premier League appearances from the bench this season, and Saido Berahino.
Berahino, 23, has managed just four league appearances this season and continues to be linked with Stoke City.
Gestede is under contract with Villa until 2020 but could be open to a move to the Hawthorns due to the lure of Premier League football.
The Baggies may face competition however, as Hull City have also been credited with an interest in the 28-year-old forward.