Trapattoni's reign as Republic of Ireland manager came to an end on Wednesday morning 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.
Former Celtic and Sunderland manager Martin O'Neill had already been installed as the bookmakers' favourite to replace the 74-year-old Italian even before his departure had been announced as it became clear his days in charge were drawing to a close.
But chief executive John Delaney insists the FAI board will not rush to make a decision as it considers its options.
Delaney told Newstalk: "The board will meet within the next week.
"Today was a day to deal with Giovanni and [assistant manager] Marco [Tardelli] in a dignified manner, and hopefully people will respect the job he has done.
"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."
O'Neill, who was sacked by Sunderland at the end of March, is not the only potential candidate to fill the vacancy, with former Ireland manager Mick McCarthy, currently at Ipswich, his captain Roy Keane and current Norwich boss Chris Hughton also figuring prominently in the betting.
Delaney said: "I think there will be plenty of interest because first of all, 24 teams go to the European Championships, it's genuinely accepted we have a good crop of young players and the monetary incentive has been good over the last number of years.
"The knowledge of the Irish job because of Trapattoni across Europe has created interest in managing the Irish team.
"It will be interesting to see who puts their best foot forward."
Asked specifically about Keane, who famously returned home from the Far East before the 2002 World Cup finals had started, Delaney said, laughing: "I don't think I will comment on any individual."
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.
"We will look at the job description over the next week.
"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.
"Sometimes in Ireland when things are great, we get too carried away; when they are not, we get too negative."