The FAI accepted 5m euros (Â£3.6m) from Fifa in 2010, in return for dropping court proceedings over Thierry Henry's handball in a 2009 World Cup play-off.
The FAI described it as an "interest-free loan" that was accounted for in its 2010 audited financial statements.
It said the 5m euros was recorded under "bank and other loans" in its accounts.
The timeline has been published on the FAI's website, and includes copies of its written agreement with Fifa and relevant bank documents.
It confirms that in 2013, the remaining loan amount was "reduced to nil as a result of the non-qualification for the World Cup South Africa in 2010 and the World Cup Brazil in 2014".
Details of the timeline were released just hours after Irish prime minister Enda Kenny publicly urged the FAI's chief executive John Delaney to explain the Fifa deal.
Mr Kenny described the payment as "quite extraordinary" and said the FAI boss must answer questions "in the interests of transparency and accountability".
The Republic of Ireland team was 16 minutes away from a penalty shootout for a place in the 2010 finals in South Africa when Henry handled in the build-up to William Gallas's extra-time equaliser in the second leg of the play-off in Paris.
In its latest statement, the FAI said that because FIFA has confirmed the transaction, FAI Board members believed they were "no longer obliged to abide by the confidentiality agreement" struck as part of the deal.
Its timeline detailed how the FAI wrote to Fifa in November 2009, saying it was considering referring the handball dispute to the Court of Arbitration for Sport.
In a press conference later that month, Fifa president Sepp Blatter joked about the FAI's request to be the 33rd team at the World Cup, which the FAI claimed caused it "reputational damage".   Mr Blatter later made a personal apology for the remark.
Following negotiations with Fifa in Zurich in January 2010, the FAI said FIFA offered it  "a 5m euros interest-free loan by way of compensation as well as a 400,000 euros (Â£290,000) Goal Project grant that was used for FAI Regional Football Centres".
It added that the 5m euros was lodged in to FAI's National Irish Bank deposit account on 20 January, 2010.
"This funding was used to make payments to New Stadium Ltd (trading name for Aviva Stadium management company) on February 25 and March 26 of 2010," the FAI statement said.
It said it had released the details "to demonstrate that the Board of the Association acted at all times in the best interests of Irish football, and in full compliance with Irish company law".

It means control over new machines with maximum stakes over £10 will be devolved, as the Wales Bill cleared its final hurdle in Parliament.
Shadow Welsh Secretary Jo Stevens wanted the power extended to existing terminals with maximum stakes over £2.
UK ministers expect the bill to gain Royal Assent in February.
The Wales Bill will hand Welsh ministers extra powers over transport, energy and electoral arrangements - including the power to increase the number of AMs.
It will also allow the Welsh Government to make some alternations to income tax.
But critics, including the Welsh Government itself, said other changes introduced too many restrictions on what it could or could not do.
On Tuesday, in the Commons, UK government amendments introduced in the Lords were accepted, devolving powers over water, fishing vessels and the right to be consulted over coastguard policy.
Welsh Secretary Alun Cairns said it was a "landmark day" and the bill "brings clarity to the settlement and accountability for Welsh Government".
In October, the UK government announced a review of FOBTs amid concern about their attraction to vulnerable people.
On the question of devolving FOBT regulation, a UK government spokesman said it was "giving Wales the same powers as Scotland has to address the local social problems that these machines can sometimes create".
Labour had claimed fixed-odds betting terminals (FOBTs) were highly addictive, allowing players to stake up to £100 every 20 seconds, compared to the £2 maximum bet on a fruit machine.
The party pointed to research showing there were four times as many betting shops in areas of high unemployment as in areas of low unemployment.
The Campaign for Fairer Gambling estimates that customers in Wales lost over £50m on FOBTs between September 2014-15.
The Association of British Bookmakers defended betting shops in Wales as the "safest place" to gamble as they were the most highly regulated retailer on the high street.
The Football Association of Ireland (FAI) has released the timeline of its controversial multi-million-euro deal with Fifa to stop a legal battle.
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A final attempt by Labour to give Wales more power to ban fixed-odds betting terminals has been defeated in the House of Commons.