The report claims that Greek side Iraklis have opted to accept Forest’s approach for the former Everton man and after talks with directors at the City Ground the player is said to be keen on a switch.
Forest are looking to strengthen their squad this summer following their exit from a transfer ban.
Despite still being manager-less, the club have made the forward options a priority after loan duo Nelson Oliviera and Ryan Mendes returned to their parent clubs at the end of this past season.
Forest have few recognised strikers to partner Britt Assombolonga in attack, though that could soon change as they edge closer to the signing of Greek forward Vellios.
The 24-year-old spent three seasons at Goodison Park after joining from Iraklis, yet a failure to break into the first-team saw him leave for Belgium and then back to Greece for a second spell in Thessaloniki.
Last season could be considered his breakthrough campaign, as he netted 11 goals in 27 league games.
Sky Sports say a bid has been accepted by the Greek club, but more importantly the player seems keen on a switch to the City Ground although it is said a decision will be made upon his return from holiday.
Forest will be keen to challenge for the Championship play-offs next season, although the club are still short of quality in certain areas and will need to make some positive additions this summer.
Everton's Magaye Gueye (L) and Apostolos Vellios celebrate win at the end of the game