Speaking on RTÉ Radio, Football Association of Ireland chief executive John Delaney said the payment related to a handball by Thierry Henry, which came in the final minutes of the match and cost Ireland a place in the 2010 World Cup.
"We felt we had a legal case against FIFA because of how the World Cup play-off hadn't worked out for us with the Henry handball," Delaney said.
He said it was a "very good" and "very legitimate agreement" for the association, and he was bound by a confidentiality agreement not to reveal the amount. The payout was $5 million (4.4 million euros).
"It was a payment to the association to not proceed with a legal case," Delaney said.
In a statement, FIFA acknowledged paying the Irish Football Association but said the money was a loan for a stadium construction.
"The terms agreed between FIFA and the FAI were that the loan would be reimbursed if Ireland qualified for the 2014 FIFA World Cup. Ireland did not so qualify," FIFA's statement read.
"Because of this, and in view of the FAI's financial situation, FIFA decided to write off the loan as per 31 December 2014."
A tumultuous few days
Right now, FIFA is the subject of two separate investigations: one in the U.S. and another in Switzerland. And each day has brought a new twist in a tangled financial tale woven by football's governing body.
Last week, U.S. prosecutors issued arrest warrants for 14 people, on charges ranging from money laundering to fraud and racketeering. They include FIFA officials who took bribes totaling more than $150 million and in return provided "lucrative media and marketing rights" to soccer tournaments as kickbacks, prosecutors say.
A separate investigation by Swiss authorities is investigating the circumstances of the 2018 and 2022 World Cup bids, which went to Russia and Qatar, respectively.
The scandal has prompted speculation over whether Russia or Qatar could lose the right to host their respective World Cups if wrongdoing is proved.
On Tuesday, Sepp Blatter -- the man at FIFA's helm -- announced he'd be stepping down, just four days after being elected president for a fifth term.
On Wednesday, court records revealed that Chuck Blazer
, a former FIFA executive committee member and a key player in the ongoing corruption investigation into international soccer's governing body, had admitted taking bribes and helped South Africa land the 2010 World Cup.
And on Thursday, former senior FIFA official Jack Warner -- who has been indicted in the U.S. investigation -- promised to reveal the secret dealings of the powerful organization, including those involving Blatter, its previously seemingly untouchable leader.