Naylor is a free agent after his contract with Sky Bet League Two club Accrington was recently cancelled by mutual consent.
The 33-year-old played the full game as the Rams' Under-21 side drew 0-0 against Sheffield Wednesday at the Owls' training ground on Tuesday afternoon.
Naylor started his career at Wolves, where he worked under academy manager Chris Evans, who is now head of football operations at Derby, and made almost 300 league appearances for Wanderers before joining Celtic in 2006.
He made 100 league appearances for the Glasgow club and then moved to Cardiff in 2010. He joined Accrington at the beginning of this season.