Wigan boss Roberto Martinez is an admirer of the England Under-21 defender who will be allowed to leave The Hawthorns on loan next week if the Baggies can get another central defender in to replace him.
However, Dawson is wanted by a number of ambitious Championship clubs who can also meet his wage package and potentially offer him assurances of regular first-team football.
Blackburn Rovers, Leeds United, Wolverhampton Wanderers, Bolton Wanderers, Barnsley, Burnley and Nottingham Forest are all in the queue with Ewood Park boss Michael Appleton potentially a step ahead of the rest because of his links to The Hawthorns.
The highly-rated 22-year-old has yet to play a league game under Steve Clarke so far this season and a number of Premier League clubs have shown an interest as a result.
But West Brom are not interested in selling Dawson just yet and have told Swansea to forget about making a permanent bid.