The Italian's reign came to an end yesterday when the two parties agreed an amicable split in the wake of Tuesday night's 1-0 World Cup qualifier defeat in Austria which effectively ended their hopes of making it to Brazil next summer.
O'Neill, who was sacked by Sunderland at the end of March and is the bookmakers' favourite, is not the only potential candidate to fill the vacancy, with former Ireland manager Mick McCarthy, currently at Ipswich, his captain Keane and current Norwich boss Chris Hughton also figuring prominently in the betting.
FAI chief executive John Delaney said: "The board will meet within the next week.
"We will meet and will determine a process. We do have time. The European qualifiers don't start until next September. We will take stock of the last couple of days and then discuss the process."
He added: "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.
"The last four or five days have been difficult for everyone in Irish football. It was important to get a decision made in a dignified way.
"We have to look at who can get the best out of the players available to us."