The Rams rejected an undisclosed bid from the Premier League newcomers for the influential 27-year-old Scotland international midfielder, who moved to the iPro Stadium from Kilmarnock in 2011 and scored 16 goals last campaign.
Clarets boss Sean Dyche was keen to get Bryson to Turf Moor to bolster his midfield options after seeing the way he helped Steve McClaren's side finish third in the Championship last term, but then miss out on promotion to QPR in the play-off final.
However, after snubbing the initial bid from Burnley, Derby are now looking for Bryson to commit himself to the club beyond the end of his current deal which expires in summer 2016.
The Rams released a statement which read: "The offer was immediately rejected and the club remains in discussions with Bryson with regards to extending his existing contract.
"Bryson, along with the rest of Derby's first-team squad, will be reporting back for pre-season training this week."