The Republic of Ireland manager revealed last week that his assistant had been contacted by Dermot Desmond, the Glasgow club's majority shareholder, about the prospect of succeeding Neil Lennon following his decision to resign last month.
A Celtic statement released soon afterwards intimated that the club were considered a host of potential candidates, with Keane featuring on a shortlist of between five and 10 figures being measured up for the vacant position. The prospect of the former Manchester United captain - who has been working as a member of Ireland's coaching team since the appointment of O'Neill in November last year - being appointed by Celtic has been the prevailing news story in his homeland, with O'Neill asked to comment on it even after Ireland drew with Italy on Saturday night.
However, with the Irish squad scheduled to fly out on Wednesday for a summer tour of the United States, O'Neill is confident that Celtic will make their intentions regarding his assistant known before then. Keane did not attend a prearranged media conference last week ahead of the friendly with Italy and is understood to have discussed the vacancy at the Scottish champions informally with Desmond.
"I think this issue will be resolved by Wednesday," said O'Neill. "My asumption is that Celtic, amongst other clubs, are looking at potential candidates. They have games coming up and they will want to make a decision fairly soon."
Should Keane - who finished his playing career with a brief stay at Celtic during the 2005/06 season - be named as manager then he is expected to resign from his post with his national team. A leading fans' group has condemned that prospect already, claiming that it will serve to imitate the controversy caused by Keane in 2002 when the midfielder walked out on Ireland at the start of the World Cup finals.
O'Neill has been more conciliatory and expressed only a hope that the 42-year-old will continue to work with the national team. "I have not spoken to Keane on this matter because we were focused on the game against Italy," said O'Neill, who managed at Celtic Park between 2000 and 2005.
"I'm really pleased with the way we played against Italy. I thought the game was very competitive from start to finish, just what both teams wanted. Italy are preparing for the World Cup, while we were looking for a boost in confidence heading into our Euro 2016 qualifying campaign and we got that.
"We would all be very disappointed if [Keane] left. My own view is that he has settled in very well indeed. The players have really taken to him, something I knew would be the case. He has been terrific. I think he is really enjoying this experience so it will be a disappointment if he left.
"The players would love him to stay. But I reiterate that this situation will be resolved before we travel to the United States."
Keane was still the favourite among the bookmakers to succeed Lennon, while the list of individuals being linked to the vacancy at Celtic has also diminshed. Former Celtic midfielder Morten Wieghorst was last week named as manager of Danish club AGF Aarhus, while Henrik Larsson has also begun to fall out of the running.
A hero among the Celtic support, the Swede has expressed an intention to remain as manager of Helsingborgs in his homeland until the end of this season. However, the former striker has also refused to deny speculation that he held clandestine talks with Celtic officials last week. A local newspaper reported that representatives from the Scottish champions boarded a private jet and set off to sound out Larsson over the vacancy, landing in Angelholm Airport, near to Helsingborg, to rendezvous with the 42-year-old.
Larsson did not deny the reports outright when given the opportunity last night. "Who told you that? I do not know anything. You can interpret that any way you want."
He was, however, more forthcoming when asked
if he wants to become the manager of Celtic later in his career. "It is a fantastic club where I became the player I was," said Larsson. "When I got into the coaching profession, I knew the manager's job would always be associated with me.
"I've seen managers going to Celtic who thought
it was just another club. But it's more than that.
I think I understand what it means to be the manager there. I definitely think I'm capable."