Keane was the bookies' favourites to replace Neil Lennon as Parkhead boss after the Irish national team assistant manager held talks with Hoops majority shareholder Dermot Desmond last week.
Sources at Celtic say Keane was not offered the job, but media reports suggest agreement could not be reached between the two parties.
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."
Former West Brom manager Steve Clarke and former Cardiff manager Malky Mackay have been installed as favourites for the post.
It has been suggested Keane turned down the chance to manage the Glasgow club to see out his commitment to the national side alongside Martin O'Neill, the former Celtic manager who is now running Ireland.
The former Manchester United captain apparently pondered the prospect of returning to Celtic, a club where he spent the final six months of his playing career in 2005/06 before retiring.
TWEET: Roy Keane would have created headlines at Celtic, but his managerial ability is somewhere between average and poor. Look for better, Celtic.
The former Sunderland and Ipswich manager Keane will head for the US this week with Ireland for games against Costa Rica and Portugal before joining ITV's team of pundits for the World Cup finals in Brazil.
However, Keane had earlier hinted it would be difficult for him to leave his current role with the Football Association of Ireland.
The ex-Sunderland and Ipswich boss - who now doubles his coaching duties with work as an ITV pundit - said on June 23: ''I've got a job - I've got two jobs - I can't have another one. They are a brilliant club, an absolutely fantastic football club, one of the best in the world. But I have got a job."
He added with a smile: ''I'm not taking a pay cut for anybody.''
- Sports & Recreation
- Neil Lennon
- Dermot Desmond
- Malky Mackay