The 50-year-old appeared to be among the frontrunners for The Hawthorns hotseat but talks have now broken down.
West Brom, who sit 15th in the Premier League and are just one point above the relegation zone, parted company with Steve Clarke on 14 December after 18 months in the job.
At least two other candidates are believed to be in the frame, although Malky Mackay, sacked on Friday by Cardiff, is unlikely to be in contention for the position.
The Baggies refused to reveal the identities of those on the shortlist but confirmed via a statement on their website: "We have identified suitable candidates to work within our existing structure.
"This is a confidential process and it is inevitable there will be speculation about who may be under consideration.
"We are now progressing through the interview and short-listing stage. We plan to make an appointment at the earliest opportunity but our primary concern is recruiting the most suitable individual.
"We are grateful for the professional manner in which Keith Downing, the coaching staff and the players have approached the last two games.
"Keith remains in charge for tomorrow's game at West Ham United and until an appointment has been made."