FIFA paid off the FAI to drop a threatened legal case over Thierry Henry's infamous handball with a $5m "loan" to help build the Aviva Stadium.
The admission came after FAI boss John Delaney confirmed that a payment was made to his organisation "not to proceed with a legal case".
The revelations further deepen the storm surrounding Fifa, which is already mired in allegations of corruption.
The FBI-led investigations and arrest of senior officials led to the resignation of its president, Sepp Blatter, though he denies any wrongdoing.
Fifa last night said that the payment to the FAI was a "loan", to be paid back if Ireland qualified for the 2014 World Cup.
It said the sum was written off on December 31 last, after Ireland failed to qualify for the tournament in Brazil and "in view of the FAI's financial situation".
Henry's blatant handball in Ireland's crucial November 2009 qualifier - missed by the referee - led to a French goal and deprived the boys in green of a World Cup spot.
At the time the FAI suggested that Fifa add Ireland to the World Cup as the 33rd team. Blatter famously publicly mocked the suggestion and the FAI later threatened legal action.
The $5m deal was struck less than two months after the game, in January 2010.
"It was a payment to the Association not to proceed with a legal case," Delaney said yesterday, though he did not reveal the figure, citing a confidentiality agreement.
Fifa released a statement after his remarks, saying that the payment was $5m, though the FAI said it was €5m.
It was described last night by Fifa as a "agreement with the FAI in order to put an end to any claims".
Its statement said that it was granted for the construction of a stadium and added that Uefa granted the FAI funds for the same stadium.
Last night the FAI said that it had "until now, abided by the confidentiality agreement required by FIFA as part of the settlement".
"The settlement was reached following strong legal advice given to the Association regarding the case against Fifa, and was a legitimate payment that enabled the Association to put €5m into the Aviva Stadium project," a spokesman said.
"This is fully reflected in our financial statements, which are audited independently.
"Furthermore the settlement was made without any conditions other than confidentiality.
"Fifa's settlement with the Association has at no time influenced the FAI's criticism of Fifa, as demonstrated by our consistent criticisms of Sepp Blatter," the spoesman said.
"It's a very good agreement for the FAI, a very legitimate agreement for the FAI," Delaney told RTE's Ray D'Arcy yesterday.
Meanwhile, Delaney said he had raised his concern over FAI officials becoming involved with Fifa's Irish company at a board meeting in October 2001.
"People like myself and Michael Cody (FAI director) expressed our concerns and we looked that the FAI would be indemnified against potential problems that might have occurred," he said.
FIFA Ireland Ltd - which had Sepp Blatter as a director - was established for tax purposes at the end of August that year and its transactions were conducted through Swiss Francs (CHF).
FAI officials Brendan Menton, Michael 'Milo' Corcoran and David Blood served as directors at various times before the firm underwent voluntary liquidation in 2008.
The Herald revealed how the firm hauled in a revenues of CHF 172.7m (approximately €113m) during its years in business from the sale of licensing rights for the 2002 World Cup.
It transferred around CHF 171m (approximately €112m) of this to FIFA in Switzerland in the form of licence fees between 2001 and 2006.
It paid CHF 291,355 (approximately €190,866) in Irish corporation tax on its profits during the same years.
FIFA said the firm was "in full compliance with all the relevant laws and regulations".
It had no directly employed staff in the period covered by its accounts and was based in the IFSC for most of its time in business.
However, the Herald can today reveal that its office moved to Bank of Ireland (BOI) headquarters on Baggot Street in 2007 and remained there until the process of winding up the company began in 2008.
BOI Treasury and International Banking was listed as FIFA Ireland Ltd's bankers in all the accounts it filed.
Company records also show that now-defunct BOI subsidiary Bank of Ireland Outsourcing Ltd acted as administrators for the company.
One of its roles was to ensure that "all receipts and payments are made in a timely manner, and thus, minimise any credit risk".
The liquidator noted "administration fees" of €15,125 paid to BOI in his Statement of Accounts.
Both FIFA and BOI declined to comment on the business relationship between FIFA Ireland and the bank.