One firm, Paddy Power, cut the price of the former West Brom and Reading manager to succeed Ronny Deila all the way down from 66/1 to 3/1.
And William Hill, sponsors of the Scottish Cup, suspended all betting on him after taking similarly evasive action.
“Some people clearly believe it is going to happen because all the bookmakers have seen money for Clarke in the last couple of days,” said Paul Petrie of leading bookmakers, McBookie.com.
“We ourselves were 40/1 and now have him 5/2, second favourite behind Davie Moyes.
“Only time will tell whether it is an informed gamble, or a reaction to some speculation. But the interest is definitely there.
“With question marks over a few of the leading contenders, the likes of Roy Keane, Brendan Rodgers and Neil Lennon, it is a very lively market.”
Clarke, out of football since being sacked by Reading in December, has been linked with several Scottish jobs.
He was in the frame to be manager of both Hibs and Rangers and, in 2014 when Ronny Deila got the job, Celtic themselves were believed to have considered him.
A boyhood fan of the Hoops, the 52-year-old missed out on the chance to join the club as a player in the late 1980s when St Mirren rejected Celtic’s bid for the defender.
He was subsequently snapped up by Chelsea, who he stayed with for 11 years, making over 300 first-team appearances.
Having gone on to coach at the end of his playing days, he was later employed in several high-profile assistant manager positions, most famously to Jose Mourinho at Chelsea for three seasons from 2004 on.
It was a hugely successful period, with the Blues winning two Premier League titles, an FA Cup and two League Cups.
He subsequently assisted Gianfranco Zola at West Ham and fellow Scot Kenny Dalglish at Liverpool before working as a manager in his own right with West Brom and Reading.
While he led to Reading to their first-ever FA Cup semi-final last season, the Royals parted company with him the following season after a disappointing start to their Championship campaign.
David Moyes 11/10.
Steve Clarke 3/1.
Neil Lennon 7/2.
Michael O’Neill 7/1.
Roy Keane 8/1.
Steve Evans 10/1.
Brendan Rodgers 14/1.
Alan Stubbs 16/1.
Roberto di Matteo 20/1.
Alex Neil 25/1.
Henrik Larsson 25/1.
Malky Mackay 25/1.
Celtic boss Ronny Deila to leave club at end of season
Celtic will win the title… but woeful signings mean Ronny Deila won’t be around to defend it
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