The Rams made the announcement on Wednesday and stated that the offer was "immediately" turned down.
The 27-year-old Scotland international's current contract with the Championship club runs until June 2016, and Derby are keen on him signing an extension.
"Derby County today received an undisclosed bid for midfielder Craig Bryson from Burnley," read a statement on the club’s official website.
"The offer was immediately rejected. The Rams remain in discussions with Bryson, who was Derby's Player of the Year in the 2013/14 campaign, with regards to extending his existing contract.
"Bryson, 27, scored 16 goals in 49 appearances for Derby last season and was named in the Sky Bet Championship's PFA Team of the Season.
"Since joining the Rams from Scottish outfit Kilmarnock in the summer of 2011, Bryson has made 134 appearances and scored 27 goals.
"Bryson, along with the rest of Derby's first-team squad, will be reporting back for pre-season training this week."