Giovanni Trapattoni's dismissal on Wednesday quickly saw Martin O'Neill installed as the bookmakers' favourite for the job.
But John Delaney has named five potential candidates, including former manager Mick McCarthy and one-time FAI nemesis Roy Keane.
He told Newstalk: "I think names like Mick, Brian McDermott, Chris Hughton, Roy Keane all come into the pot - Martin O'Neill, of course.
"I wouldn't want to comment on any individual chances. Mick has done a very good job for Ireland and in England as well. We will look at the job description over the next week."
Delaney is determined to take his time over appointing Trapattoni's successor, and the successful candidate will be given the brief of getting Ireland's "good young players" to a major tournament.
"I think he [Trapattoni] felt he had done a good job," Delaney said. "When he first came, we hadn't qualified for a major tournament for a long time.
"He genuinely believes there's a group of young players available to the next manager. Most people in Irish football would agree he's left Irish football in a better state."
One young player given a chance under Trapattoni was James McCarthy, and the Everton midfielder would like to see Keane get the job.
The former captain was critical of the FAI during his time as a player, famously walking out days before the 2002 World Cup after complaining about the pre-tournament training camp.
But McCarthy said: "It would be great to work under Roy Keane but I'm not sure what's going to happen. It's up to the FAI and I'm sure whoever comes in will do a good job."