Republic of Ireland assistant manager Keane was thought to be close to being handed the job but surprisingly asked the Scottish champions not to consider his candidacy.
That now leaves Clarke, 50, as the bookies' odds-on favourite, although there is competition from former Cardiff boss Malky Mackay and Owen Coyle, who has not worked since being axed by Wigan last year.
He was officially relieved of his duties five months later, despite having led the club to eighth place in the Barclays Premier League in his first season in charge.
The Hawthorns post had been his first as a manager in his own right having worked as a number two at Newcastle, Chelsea, West Ham and Liverpool.
Keane dropped out of the Hoops selection process after deciding he would prefer to continue his work alongside Martin O'Neill.
Celtic's majority shareholder Dermot Desmond held talks with former the Manchester United skipper last week but it is understood the discussions did not get as far as a formal job offer.
A source close to the club told Press Association Sport: "Roy was not offered the job - no-one has been offered it yet.
"But he has ruled himself out and asked not to be considered for the position."
Keane will now travel with the Irish squad to the United States ahead of friendlies with Costa Rica in Philadelphia on June 6 and Portugal in New Jersey on June 10.
In a statement, the Football Association of Ireland confirmed Keane would be staying put.
It read: "The Football Association of Ireland today (June 2) confirmed that Roy Keane will travel with the Republic of Ireland squad to the US after he decided not to pursue the informal talks that took place with Celtic any further.
"The Republic of Ireland squad will gather again in Dublin on June 3 before travelling to New York on June 4."
Keane has been linked in several reports on Tuesday with becoming assistant manager to Paul Lambert, another former Celtic player, at Aston Villa.
Lambert is seeking a new number two after his former deputy Ian Culverhouse was sacked last month due to disciplinary reasons.
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- Roy Keane
- Malky Mackay