O'Neill has only won five of his 15 matches in charge of Forest, one of the clubs for whom he starred as a player, and lost his last three.
As a result the Reds, who were seventh at the time of Aitor Karanka's January sacking, are eight points adrift of the top six with four games remaining and almost certain to remain a Championship club next season.
And Sportime claims this has led Evangelos Marinakis, the Greek owner of Nottingham Forest, to open the search for a replacement.
It adds that a number of candidates have already emerged, many of whom are known to "the patriotic Greek territories".
O'Neill is Forest's 12th permanent manager in 10 years.
Who do Nottingham Forest fans want as the next City Ground boss?