Martin expects Roy Keane's future to be resolved before the Republic of Ireland team flies to America on Wednesday.
Despite Celtic CEO Peter Lawwell's claim that the SPL champions are considering up to 10 candidates, the Irish camp are bracing themselves for the departure of their high-profile assistant boss.
Lawwell did confirm that Keane was in the frame and described the hot favourite as a "first-class guy."
The Cork man had informal talks with Celtic's majority shareholder Dermot Desmond earlier this week and it is anticipated that matters will progress after the Republic's friendly with Italy at Craven Cottage tonight (7.45pm).
Keane flew with the Republic squad to London yesterday, but O'Neill was unable to say for certain that the 42-year-old would travel to the US on Wednesday for the games with Costa Rica and Portugal that bring down the curtain on the summer gathering.
The Republic manager believes there will be clarity on the dominant issue prior to departure.
"I would imagine that in the next few days that the situation will resolve itself," said O'Neill.
"Whether it goes into Monday, Tuesday, Wednesday. We are not actually leaving until Wednesday, so I would imagine the situation would be sorted by then.
"Celtic are actively seeking a manager and they have a certain number of days to try and do something, because they will want a man in place for when they come back for pre-season.
"There's nothing definite yet, but I can imagine that before we travel to America something will have happened.
"He still has to be asked about the job. We would all be very, very sorry to see him go if, indeed, he does."
Stand-in skipper John O'Shea does not want to lose Keane, but acknowledged the attraction of Celtic.
"If there were ever any team that could entice him back into a management role, I'm fairly certain Celtic were up there," asserted the Waterford man.
"It seems to be very close. We always knew there was a threat of losing him and Celtic seem to have fell into place for him at right time."
O'Shea also believes it will be a swift process.
"Everyone involved is too experienced to let it drag on," he continued.
"Everyone will want it done and dusted soon. If he takes the job, he's got a game fairly soon."
'Silence is golden' is Keane's motto as the speculation gathers pace.
O'Shea says the Irish players have been trying to get Keane to spill the beans on his future, but to no avail.
This is the 'new' Roy Keane, turning 43 in August, and with a mixed bag of experiences in club management on his CV.
Just over nine years ago, the 'old' Keano let rip in an infamous MUTV interview about the attitude and performances of team-mates at Old Trafford.
United chiefs banned the screening of the interview, but the incident set in train a series of events which culminated with Keane's abrupt departure in November 2005, and paved the way for his brief stint as a Celtic player.
O'Shea experienced the turmoil of the end of the Keane era as a United player, and if the bookmakers and pundits are any judge, he's going to witness another unexpected Keane departure to Glasgow, filling the seat vacated by Neil Lennon.
O'Shea added: "The lads have been trying to get a few hints, but he has kept a good poker face.
"The lads are obviously talking about it, don't get me wrong. The lads are trying to pick up things. Is he on the phone? Is he not on the phone?
"With the information around with Twitter or whatever, Sky Sports News on in the hotel 24/7, speculation is not only in the papers, it's in the hotel too.
"As the manager said, we'd be very disappointed if he leaves, because we know what a fantastic character he is to have fighting for us and on our side going into matches."
* Celtic defenders Emilio Izaguirre and Mikael Lustig have both signed new three-year contracts.