The Sunday Mirror says Birmingham's Chris Hughton - interviewed at the time Hodgson took over from Roberto Di Matteo - is a hot favourite for the job, with former first team coach Michael Appleton, who left to boss Portsmouth, also in the running.
Baggies defender Gareth McAuley remarked: "I'm sure the club will be doing things behind the scenes. They know the last couple of guys who have been in charge have taken the club forward and they will want to progress again.
"The sooner they do that, the better it will be for us as a club to move on. The new guy will have a chance to properly assess the squad and look at what he wants to do."