The Scotland international, 30, joined Cardiff from Norwich City in 2009 and has gone on to make 237 appearances during his six years with the club.
Baggies boss Tony Pulis is looking to bolster his goalkeeping options ahead of the new Premier League season, with No 1 Ben Foster not expected to return from a cruciate knee ligament injury until October at the earliest.
Boaz Myhill and youngster Jack Rose are currently the only other goalkeepers at Pulis’ disposal.
Marshall, who has three years left on his current contract at the Cardiff City Stadium, helped Cardiff win promotion to the Premier League in 2013 and was an outstanding performer for them during the following season in the top flight.
Despite producing a number of superb performances, Marshall was unable to prevent Cardiff from being relegated back to the Championship.
Pulis has already brought James McClean, James Chester and Rickie Lambert to The Hawthorns this summer.