21-year-old Bulgarian goalkeeper Evtimov has put pen to paper on a four-year deal and will presumably be loaned out after the signing of Leicester's Ben Hamer on a season-long loan recently.
Winger Burke, 18, is now a fully-fledged professional with a three-year contract and will look to build on his three senior appearances from last season.
Meanwhile, the Nottingham Post reports that of the two trialist strikers who have spent pre-season with the Reds, only Paul Taylor remains in with a chance of staying at the club.
The lack of space in Dougie Freedman's squad means no deal for Nicky Maynard although defender Daniel Pinillos does look set for a longer stay in Nottingham.