Pards moved to complete the unexpected coup as West Brom hurriedly set up Sturridge's medical - while his former club Newcastle failed to get out of the starting blocks.
Newcastle submitted two offers for the Denmark global but Feyenoord are refusing to lower their £20m asking price.
The move comes with Newcastle's pursuit of Feyenoord's Nicolai Jorgensen having hit an impasse over the two clubs' respective valuations of the player.
However, West Brom have emerged as favourites for the 28-year-old, who is looking to play regular football away from Anfield between now and the summer's World Cup finals.
Newcastle, meanwhile, now sit 15th in the Premier League and just a point above the relegation places.