It is understood the Baggies have refused the 24-year-old's request as new manager Alan Irvine is currently assessing the squad he inherited from out-going boss Pepe Mel.
He is the second West Brom player to be denied a move after George Thorne, who spent last season on loan at Watford and Derby, received a similar response.
Dawson has recently been linked with a move to newly-promoted Burnley while the arrival of Joleon Lescott at the Hawthorns will do little to enhance his chances of first-team involvement next season.
The Rochdale-born player, who joined from his home-town club in August 2010, made 15 senior appearances last season.
However, for much of the campaign, he was mainly viewed as a back-up to Gareth McAuley and Jonas Olsson.
Albion striker Victor Anichebe, meanwhile, has warned the club's youngsters to be ready to take their chance as Irvine prepares to start work in earnest.
Anichebe emerged from the ranks at Everton during Irvine's time as assistant to David Moyes, and is confident he will look to youth in his new role.
He told the club's official website: "The players will enjoy great training sessions, they will enjoy the way he plays and they will enjoy his character as a person, especially the younger players.
"They are the ones who are going to be the future, and we have got a few really good, exciting young players. He will give them a chance if they are up for it and they are eager.
"I think that's something Albion fans should look forward to."