Keane is one of the candidates the Scottish champions are considering as they seek to replace Neil Lennon, who resigned as at the end of the season after four years in charge.
"I have not spoken to Keane on this matter because we were focused on the game against Italy," O'Neill said after his side's goalless friendly draw with the Italians at Craven Cottage on Saturday night.
"I think this issue will be resolved by Wednesday.
"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."
Keane, who finished his playing career with Celtic, has made a huge impact on the Irish national side since becoming O'Neill's assistant in November.
"We would all be very disappointed if he left," O'Neill said. "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."
Ireland head to their summer tour in buoyant mood after an impressive display against Italy.
The Irish put in an improved performance against the Azzurri, following on from their 2-1 defeat against Turkey in Dublin last weekend.
They created numerous chances and were unlucky not to come away with a win after Stephen Quinn's effort hit the crossbar with 12 minutes remaining.
"I thought we were excellent," O'Neill said. "We could have scored a few goals.
"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.
"I think in the recent games that we have played, apart from the one against Poland (a goalless draw in November), we probably should have won all of them considering the scoring chances we've created.
"I hope we are saving those goals for our European qualifying campaign."
Ireland will have Robbie Keane available for the friendlies against Costa Rica on June 7 in Boston and against Portugal three days later in New York.
Keane, who is the country's all-time scorer, is playing with club side Los Angeles Galaxy.
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- Neil Lennon
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