Premier League new boys Burnley have made Dawson one of their top targets this summer and have already seen two bids rebuffed for the centre-back.
Burnley saw their first offer rejected in June before coming back with an improved bid last week which was also turned down by West Brom.
However, Sean Dyche's side are determined to land former England Under 21 international and are believed to be readying a third offer for the talented defender.
West Brom are reluctant to lose Dawson as they want to keep hold of their best players, but could risk seeing him becoming unsettled if he does not play regularly in Alan Irvine's starting XI next season.
Dawson has found Premier League action hard to come by since moving to The Hawthorns from Rochdale, and Burnley are hoping the chance to play on a regular basis at Turf Moor could help lure the 24-year-old away from the Hawthorns.