Outgoing Fifa vice-president Jim Boyce has called for an investigation into the €5m (£3.6m) given to the FAI to not contest Thierry Henry's infamous World Cup handball.
As a tumultuous week at Fifa continued, FAI chief executive John Delaney said the organisation had reached an agreement with president Sepp Blatter to not take the case to the courts.
Delaney also insisted that the millions handed over for the construction of the Aviva Stadium were legitimate and a significant boost.
"It's a very good agreement for the FAI, a very legitimate agreement for the FAI," he added. "We felt we had a legal case against Fifa because of how the World Cup hadn't worked out for us with the Henry handball."
The FAI said it accepted Fifa's settlement to avoid a "long, costly legal case".
Fifa last night said the payment to the FAI was a "loan" to be paid back if Ireland qualified for the 2014 World Cup. It added the sum was written off after Ireland failed to qualify for the tournament in Brazil.
But Mr Boyce, who stepped down as Britain's Fifa vice-president last week, was dumbfounded by the revelations.
"How anyone can authorise a payment of €5m to stop legal proceedings is totally and utterly beyond me," he said.
"I've never known in my experience of football of someone giving a payment like that for a referee's mistake to stop someone taking legal action."
Adding more fuel to the fire of the Fifa corruption crisis, the Belfast man and former IFA chief labelled the payment "ridiculous" and said there should be a full investigation into such "arbitrary" payments.
Boyce said: "I'm absolutely astounded - I have never heard anything as ridiculous in my life. If a payment of five million has been paid because of a handball and threatened legal action, then I hope a full investigation will be carried out into this and any other such arbitrary payments."
Fifa said in a statement: "While the referee's decision is final, and the FAI ultimately accepted it as such, in January 2010 Fifa entered into an agreement with FAI in order to put an end to any claims against Fifa."
The development focuses further attention on Fifa's unorthodox business practices after the disclosure this week it paid disgraced Fifa vice-president Jack Warner on behalf of South Africa ahead of the 2010 World Cup.
A leaked letter from the South African Football Association showed they instructed Fifa secretary general Jerome Valcke to pay $10m (£6m) to Warner in 2008.
Within hours of the letter being made public, Sepp Blatter had announced he would stand down as Fifa president as soon as a new election takes place next year.
However, there have been calls for Blatter to leave immediately, with Warner describing him as a "lame duck president'', and promising an "avalanche" of information to bury the man he helped get to power.
"You can't say you're stepping down and then stay for several more months,'' Warner said. "You can't do that. You will be a lame duck president.''