HITC Sport understands that Cardiff goalkeeper is on his way to The Hawthorns as Tony Pulis wants competition between the sticks.
30-year-old Marshall has been linked with a host of Premier League clubs since Cardiff City's relegation in 2014 - but he stayed with the Bluebirds last season.
Sources have confirmed to HITC Sport that West Brom are hoping to have Marshall signed and sealed by Monday, ahead of the start of the new Premier League season.
Marshall has three years left on his current deal, but he is set to sign for West Brom where he will compete with Boaz Myhill to start the Premier League season against Manchester City on May 10th.
West Brom's long-term number one Ben Foster is still sidelined with a knee injury and is not expected to be fully fit until late in the year.
Should Marshall finalise his move, he will become West Brom's fourth signing of the summer as Tony Pulis has already brought in James McClean, James Chester and Rickie Lambert this summer.
Marshall will end a six-year spell with Cardiff following a move from Norwich City, which has seen him play over 200 games for the South Wales club.