Campbell is looking for a new club after sensationally walking out on League Two side Notts County less than a month after joining them.
Reports suggest the former Tottenham and Arsenal star has opened preliminary talks with Albion about a possible move to Robertyo Di Matteo's high-flying Championship outfit.
A number of clubs, including Hull City, Ipswich and Newcastle, have also been credited with an interest in Campbell.
But it appears the Baggies are hoping to steal a march on their rivals by stepping up their interest in the 35-year-old.
As things stand, Campbell is not allowed to join a new club until January, but the former England international is hoping to be given special dispensation to find new employers before that.
Campbell said earlier this week: "There are ways of playing before January; I'm not going to elaborate, but there are ways.
Not very soon, but there are ways.
"First of all, I've got to get out of the (County) contract, people have to sign papers. There are lots of things that have to be tied up but I think everyone understands why."
County chairman Peter Trembling told Sky Sports News on Thursday: "Sol didn't want to play for Notts County, he wanted to go and play elsewhere - of course he can't do that until January but I wish him all the best.
"We've just got to make sure everything's fine in respect of what is best for the club and what is best for Sol. We're talking about that and we'll hopefully meet on mutual ground and get it sorted."
Questions were raised over the centre-half's fitness after his disappointing showing at Christie Park, while his County salary - a reported £50,000 a week - could also prove a sticking point for any possible move.