Burnley have previously expressed their interest in signing both players and it was reported that they made a combined £25m bid for the pair back in June that was rejected by West Brom.
The Clarets then made a revised £15m bid for Rodriguez, but that was also rejected by the Baggies, who are now demanding £20m for each player.
Dawson has already pushed for a transfer elsewhere and has handed in a transfer request whilst refusing to travel to the club's pre-season training camp in Portugal. Rodriguez has not handed in any official request, but it would come as no surprise if he was unwilling to play in the Championship next season.
According to Sky Sports, West Brom will be willing to part ways with both players, but only for the right price. Since setting the combined price tag of £40m, there has been no contact between the clubs.
If West Brom are able to convince Burnley to cough up then they would be earning a sizeable profit on both players. Rodriguez signed for the Baggies last summer for a fee said to be worth around £12m and Dawson joined from Rochdale in 2010 for £175k.
Burnley will be keen to keep negotiating with West Brom as they don't have long left to bolster their squad for the upcoming season. Should they get past Aberdeen in their Europa League qualifier, they will be in for a long season and will need as many capable players at their disposal as possible.