Striker Rodriguez scored 11 goals last season for the Baggies despite a difficult campaign that culminated in relegation from the Premier League.
Defender Dawson was also heavily involved in the 2017-18 campaign, making 28 league appearances for West Brom.
According to Sky Sports News, Burnley made a renewed offer in the region of £15m for Rodriguez last week but it was not deemed acceptable by Albion, who want at least £20m for each player.
The report also claims that the two clubs have not had any contact since following West Brom's rebuttal.
Dawson has subsequently handed in an official transfer request to the WBA board and refused to go on the club's training camp to Portugal, while Rodriguez did travel.