Betting on the Cork native taking over at Parkhead was slashed on Thursday and Republic of Ireland manager Martin O'Neill has confirmed his assistant has spoken for the SPL side
Neil Lennon decided to quit the Scottish champions after guiding them to a third consecutive title in his fourth season.
Keane spent a six-month spell at Celtic as a player before retiring in 2006, and O'Neill has revealed the 42-year-old has held "informal talks" about returning to succeed Lennon.
"I had a call from Dermot Desmond asking me to talk to Roy [Keane] if it was possible," said O’Neill. "I mentioned it to Roy. I am sure there were talks, informal talks.
"I would be really disappointed if he leaves. But, it kind of felt when I took him on in the first place that over the course of two years, that this would happen.
"But I did not think it would happen so quickly. He has had a number of offers from a number of football clubs. He has a lot to consider.
"Celtic are a very big club. Particular offers like this won’t come around too often. It is a very big job, obviously."
O’Neill is certain of the fact that Keane will be with the squad for the upcoming friendly against Italy but cannot guarantee his presence in America where Ireland play Costa Rica and Portugal.
"He certainly would be involved on Saturday," said O’Neill. "Of course he will keep me informed the whole way through.
"We will take it step by step at the moment. Things can develop really quickly or really slowly. I would be disappointed if Roy does go, obviously.”"
Asked if he had any regrets in bringing Keane into the Ireland set up, he said: "Absolutely none, and he is not gone yet.
"We are at a very speculative time at the moment. I think with Roy [Keane] coming into this job, that these opportunities would come about. We will all be disappointed, the players, the backroom staff. Everyone."