His answer will be given before the Republic of Ireland squad flies to Philadelphia for the first of two friendlies in the States.
Keane has already told former Hoops star and current Republic international, Aiden McGeady, that, if he takes the job, he will most likely stay in Edinburgh, as he did when he spent five months playing for the club in 2006.
But, although the 42-year-old – who is understood to have spoken to the FAI about the possibility of combining his position as assistant to Martin O’Neill with the national team and managing Celtic – is very much in pole position to succeed Neil Lennon, he is not the only runner still in the race.
Henrik Larsson – who was to the fore when the club started their search for a replacement manager, but who appeared to rule himself out on the grounds it was the wrong time for him – has jumped back into the picture after a weekend of intrigue.
The man who is under contract at Swedish club, Falkenberg, until November, was prickly when asked if it was true a deputation from Celtic had flown in for talks with him last week.
But he did go on to say: “I’ve seen managers going to Celtic who thought that it was just another club – but it’s more than that.
“I think I understand what it means to be the manager there and what is expected of you – and I definitely think I’m capable.”
Another strong candidate from the outset, David Moyes, admits he has received tempting offers.
And, though he is considering taking some time out of football after being sacked by Manchester United, the former Hoops defender has not ruled out a quick return to the game.
Celtic’s major shareholder, Dermot Desmond, is a big admirer of the 51-year-old, who made such a success of managing Everton on a player budget which was considerably smaller than that of clubs who regularly finished below them in the Premier League.
Chief executive Peter Lawwell insisted they have up to 10 “exceptional candidates”, and Moyes would certainly tick that box, along with Malky Mackay, Owen Coyle, Steve Clarke and Michael Laudrup.
Whoever becomes the new boss is expected to be faced with a decision over the future of Beram Kayal, who is entering the final year of his contract and is set to be the subject of a bid from Maccabi Haifa.