The 42-year-old has been identified as a leading candidate to become the new Celtic manager after discussing the vacancy with the club's majority shareholder, Dermot Desmond.
Returning to the club where he had a brief spell at the end of his playing career would force Keane to resign from his position with his national team, where he works alongside Martin O'Neill, the former Celtic manager. Keane was given permission to sidestep a prearranged media briefing ahead of Ireland's friendly with Italy tonight, partly as a result of the speculation surrounding his future. Instead O'Neill confirmed the approach from Celtic and it is likely that he did so with the consent of Desmond.
Some Ireland supporters have been open, too, in condemning the idea of Keane electing to abandon his post with the national team. Sections of the Irish support still remember clearly when the former midfielder - who won 67 caps for his country - walked out on the squad prior to the 2002 World Cup finals, unwilling to accept what he perceived to be inadequate training facilities in Saipan and insufficient planning from the Football Association of Ireland.
It was an episode which caused divisions in the camp and right across Ireland, and one fans' representative last night acknowledged that Keane's proposed return to Celtic Park would reopen those wounds.
"It would be wrong. It's Saipan Two - he's only in and he's out," said Gerry Rochford, chairman of the Ireland Supporters Club in Dublin. "I was at Saipan. It was a political thing at the time and I think deep down a lot of people haven't forgiven him.
"The country didn't come before him, he came before the country, and this could maybe be something like that. The aim was to get us to [the European Championships in] 2016 and to walk out the door so quickly, I don't think fans will forgive him for that. For him to walk out now, he's not coming back if he saw Ireland as a stepping stone just to get a job."
Desmond may be unlikely to have made an approach only to offer the job of Celtic manager to somebody else, although the club made an effort last night to insist that their one-time midfielder was but one name on a list of between five and 10.
"We are going through a process," said Peter Lawwell, the Celtic chief executive who also confirmed that his club has re-signed defenders Emilio Izaguirre and Mikael Lustig on three-year contracts.
"Hopefully it won't take too long but I'm sure you'll understand there's not much more we can say when we are midway through that process. There is no point in getting it done quickly and getting it wrong."
While there is voluble opposition to his appointment in his homeland, there was last night a differing view from within Scottish football. Paul Hartley, the former Hearts midfielder, faced Keane in league competition and acknowledged that the Irishman has the personality necessary to handle the pressure of the role. "Celtic need a strong character just like Neil Lennon and Keane fits that bill," said the Dundee manager.