The 19-year-old England youth international has agreed a three-year contract at Pride Park and will initially join the club's under-23s.
Duncan left Liverpool in acrimonious circumstances last year after his agent accused the club of 'mentally bullying' him - a claim the Reds denied.
He joined Fiorentina for £1.8m soon after and played in their academy.
Last December, Duncan's agent, Saif Rubie, was suspended by the Football Association for six weeks and fined £10,000 for "improper" comments on Twitter - he had also accused the then-European champions of keeping Duncan at the club "against his will", a claim Liverpool also denied.
"Bobby Duncan is a player with plenty of potential and we are delighted that he has chosen to join Derby County," the Rams' academy director Darren Wassall told the club website.
"Initially, Bobby will join our under-23s squad. Like with any other player in our academy set-up, if they perform both in training and in matches then there is a pathway at this club to progress to the first team."
Writing on Instagram, Duncan said: "I am delighted to have signed for Derby County Football Club.
"I took the move to Italy with only one thing as a priority, to try and develop further as a [f]ootballer.
"The way in which the last transfer was facilitated was incorrect & unnecessary and has all proved to be a very big learning experience for me as a player and as a person.
"My intent has always been to develop. Now I have returned to England with an accumulation of experiences under my belt and am without doubt a better player and stronger person for it.
"It is time for my [f]ootball to do the talking. I am very excited by the opportunity presented to me at Derby County and will give absolutely everything to the club moving forward."