Freed by West Bromwich Albion at the end of last season, attacking midfielder Barnes was touted as a hot property when coming up through the ranks at Derby.
Injuries have since taken their toll on the 22-year-old but Steve McClaren is believed to be interest in a deal as he reshapes his new squad.
Then West Brom manager Roberto Di Matteo signed Barnes in February 2010, saying at the time: "There is no question that if Giles can get his career back on track, we'll have a very talented player on our hands."
Other names in the frame for a move to the City Ground include Blackburn Rovers midfielder Keith Andrews, former Forest favourite Andy Reid and Bristol City striker Nicky Maynard while McClaren is keen to keep soon-to-be free agents Robert Earnshaw and Guy Moussi.
He told BBC Radio Nottingham: "I don't want to lose good players. Robert and Guy are very good players.
"I've got an opportunity to convince them Forest is the club to be at."