However, Sky sources understand that no offer has been made to Keane yet.
Keane is currently O’Neill’s assistant with the Republic of Ireland, who are preparing for a friendly against Italy on Saturday.
But TV3 reporter Tommy Martin told Sky Sports News that Keane did not appear as scheduled for a meeting with the press at the Aviva Stadium on Thursday after an open training session.
O’Neill arrived instead and revealed that (Celtic majority shareholder) Dermot Desmond contacted him several days ago asking for permission to speak to Keane about the job.
“Roy has a lot to consider,” said O’Neill. “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.
“Of course I would be really disappointed if Roy was to leave us.”
O’Neil, who spent five years as Celtic manager himself, believes Keane’s loss would be a blow for Ireland but insists he always expected clubs to come knocking for his number two.
“I only reiterate what I said right at the beginning, which I think has been enhanced over this last period of time,” he added.
“Roy has been, selfishly, he’s been great for me. I’ve genuinely enjoyed working with him.
“He’s sometimes a totally different character to the one that you see on the TV. He’s been excellent, the players have enjoyed him immensely.
“I can only say that naturally I’d be disappointed if he left because I really think that he’s been terrific.
“We have only had four games with the squad, but he’s been excellent and I don’t think that he’d give that up lightly.
“Even just let’s say that it didn’t materialise – is that to say that in say October-time, the same situation might not arise?
“You have to be ready for these things. I really thought when I took him on that the period of time that we would have, at some stage or another, he would be offered something.
“I just probably didn’t expect things to happen so quickly.”
Sky Sports reporter Charles Paterson said on Thursday evening: “This is very big news.
“Celtic are saying nothing on the record but I can tell you from speaking to sources close to the club tonight that Roy Keane has not, as yet, been offered the job of Celtic manager.
“It is my understanding that Celtic are still considering a number of high-profile candidates.
“But Roy Keane has certainly been regarded as favourite over the last 24 hours and now we have confirmation that there has been an approach from Dermot Desmond.
“It is really Dermot Desmond who is driving this. Roy Keane finished his career at Celtic and that move was heavily brought about by Dermot Desmond.
“A lot of the Celtic fans we have spoken to said they would like to see (former Cardiff boss) Malky Mackay given the job but I understand Malky Mackay’s representatives have so far had no contact with Celtic.”