Marinakis axed Warburton as a result of their inconsistent form this term.
Forest sit 14th in the Championship but have lost 14 of their 26 league games.
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Barnsley boss Paul Heckingbottom is the bookies' favourite
Marinakis believes the team are underperforming and should be challenging for a play-off spot.
Academy chief Gary Brazil has been in caretaker charge since Warburton’s sacking.
He presided over a goalless draw at Leeds United on New Year’s Day before masterminding the famous FA Cup victory against Arsenal.
Brazil has indicated he does not fancy the job on a full-time basis.
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