Forest owner Al Hasawi has told BBC Nottingham Sport that prospective new owner Marinakis is meeting with the English Football League this week.
A previous takeover by a United States consortium, led by businessman John Jay Moores, fell through in January.
The Reds ensured their Championship survival with a 3-0 win over Ipswich Town on the final day of the season.
They only stayed up on goal difference and manager Mark Warburton said the club should "never be in that position again".
He would not be drawn on the likely takeover by the controversial figure of Marinakis, but said there was plenty of reason for optimism at the City Ground.
"A relegation dogfight is not what we are about," added Warburton. "I am confident it won't happen next season, and I say that because I have seen the quality within the squad.
"We need a tight squad of 21 or 22 players next season. I have got good young players coming through to fall back on too.
"Now is an important time for the club. It is all about support network.
"I am sure the guys well above my pay grade will get that work [takeover] done. It is nothing to do with me."
Colin Fray, BBC Radio Nottingham
"It is not a time for celebration, it's a time for relief. But it is also a time for looking forward and for making sure this is Forest's lowest point.
"The quicker the takeover goes through the sooner Mark Warburton and director of football Frank McParland can begin the rebuild.
"It's a been a shambolic season and one Forest just want to forget."
The deal to sell Nottingham Forest to Olympiakos owner Evangelos Marinakis is "99%" complete, says Fawaz Al Hasawi.