The former Everton striker could be City Ground bound.
Vellios first played in England with Everton
And it is claimed the offer will likely be accepted by the Super League side, despite reported interest from Olympiacos, AEK Athens and Mainz 05.
Vellios, who spent three years on Everton's books, scored 13 goals in 32 appearances for Iraklis last season, but only has a year to run on his contract.
Vellios celebrates after scoring for the Toffees
Forest head into the summer a little light in attack, with last season's loan pair Nelson Oliveira and Chris O'Grady returning to Benfica and Brighton & Hove Albion respectively.
The Championship side recently had their transfer embargo lifted, but remain without a manager three months on from the sacking of Dougie Freedman.
A proposed takeover by Olympiacos chairman Evangelos Marinakis is also still to be resolved, although Forest's Vellios pursuit, if true, suggests possible Greek involvement.
Marinakis has been linked with a takeover at Forest
Vellios, 24, began his career with Iraklis and had spells in Blackpool, Belgium and Denmark before returning to his homeland last summer.
Freedman was sacked in March