The Premier League new boys made initial contact in early June, with a £1million approach knocked back.
Dawson then saw a transfer request turned down by West Brom as new boss Alan Irvine looks to weigh up all of the options available to him.
It could be that a change in the dugout plays into Dawson’s hands as he has found top flight football hard to come by since moving to The Hawthorns from Rochdale.
A successful loan spell at Bolton Wanderers in 2012/13 allowed him to show what he is capable of, but he was restricted to just 15 outings in all competitions last season.
Burnley are willing to offer the 24-year-old the opportunities he craves as they prepare for life back in the big time, but their second offer has been snubbed and they will have to rethink their transfer strategy if Dawson is to become the latest fresh face to join the ranks at Turf Moor.