Byrne spent a season-long loan with Rushden & Diamonds, scoring seven goals for the Blue Square Premier side.
The 21-year-old has been limited to substitute appearances in the Forest first-team.
Young player of the year Karlton Watson has also been offered a new deal, as have Tom Mullen, Zannettos Mytides and Kieron Freeman.
Forest have also announced they are releasing Shane Redmond, Jordan Fairclough, Jean Biansumba, Danny Elliott, Tim Hopkinson, Sean McCashin, Carl Sibson, Matthew Sykes, George Thompson, Max Wright and Tony Diagne.
Source: BBC Sport
Source: BBC Sport