The newly-promoted Clarets are believed to have had a £1m offer for the former England Under-21 defender rejected by the Baggies at the start of the summer.
Dawson, who has made just 17 Premier League starts since arriving at The Hawthorns in 2010, has reportedly lodged a transfer request but that was turned down by new Albion boss Alan Irvine.
It seems that the managerial change at West Brom may improve Dawson's first-team chances, with the Midlands outfit having knocked back an improved bid from Burnley, according to Sky Sports News.
The 24-year-old is desperate to play regular football, and Sean Dyche is prepared to offer him that at Turf Moor. Joleon Lescott, Jonas Olsson and Gareth McAuley stand in his way at the Baggies.