Republic of Ireland assistant manager Keane had been the favourite to take over from Neil Lennon but he has turned down the chance to take over at Celtic Park to continue working alongside Martin O'Neill.
Dermot Desmond, Celtic's majority shareholder, held talks with the former Manchester United captain last week but it is understood the discussions did not get as far as a formal job offer.
A source close to the club said: "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."
The Football Association Of Ireland released a short statement confirming that Keane would be staying as part of their coaching team.
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."
Keane's departure from the scene now leaves former West Brom manager Steve Clarke as the new favourite to land the job.
Sacked Cardiff City boss Malky Mackay and Owen Coyle - who has been out of work since leaving Wigan last year - are also in the running.
Keane's U-turn comes as a surprise after reports on Monday had suggested he was within 24 hours of taking the Celtic job.
O'Neill confirmed on Thursday evening that his 42-year-old assistant had held informal talks with the Hoops, for whom he played briefly at the end of his career, after Desmond had made contact.
O'Neill said: "I had a call the other day from Dermot Desmond wanting to speak to Roy if it was at all possible, and I mentioned it to Roy, of course.
"I am sure he was wanting to have a chat anyway because a club like Celtic might not come around too often.
"It was just a really informal chat, I think, but if you are asking me would I be really disappointed? Of course if Roy was to leave us."