Warburton has been axed by City Ground bosses after nine months in charge.
Forest owner Evangelos Marinakis is expected to waste little time in appointing a new manager.
Paddy Power have installed Karanka as the leading contender at 3/1 to fill the vacancy.
He guided Middlesbrough to Championship promotion in 2016.
Former England boss Steve McClaren and ex-Leeds and Boro gaffer Garry Monk are both 6/1 shots.
Forest academy boss Gary Brazil, who has taken caretaker charge, is also 6/1 to be given the job on a permanent basis.
Warburton was given the boot in the wake of yesterday’s 1-0 home defeat to Sunderland.
His assistant David Weir and director of football Frank McParland have also left the club.
A statement read: “The club would like to thank Mark, David and Frank for all their efforts during their time at The City Ground.”
Warburton, who was attempting to change the team’s style of football with one of the Championship’s youngest squads at his disposal, pleaded for time in the wake of the Sunderland loss.
“The one commodity managers need is time,” he said. “I use Lee Johnson as the example because he had a tough run. He had eight straight defeats and one win in 19 at one stage last season. Bristol City stuck with him and look where they are now.
“Mick McCarthy was getting booed off by his own fans at Ipswich at this point last season, but look at them now.
“Neil Warnock was given time at Cardiff since last season and look at them now.
“I got angry with the media the other day when they talked about the pressure being on now.
“I read the reports and it is almost a self-fulfilling cycle. Because the media say you are under pressure, then they ask the fans to vote whether you would want to see a manager stay or go.
“It is a constant cycle. Managers need time. If I do not get time – I hope very much that I do – there is still enough quality there in that squad that, in 18 months, it will show these fans how good they can be.”