FAI chief executive John Delaney will not be called to answer questions in front of an Oireachtas committee.
There had been suggestions that Delaney would be invited to the Dáil Transport and Sport committee to discuss the €5million payment from FIFA made in 2010 following Thierry Henry's handball in the World Cup play-off with France in November 2009.
However the committee has now decided the matter does not fall under its remit, as no State funds were involved in the payment.
Labour Party TD Sean Kenny tweeted: "Dail Oireachtas Transport & Sports Cttee decided by majority vote not to invite John Delaney, FAI, before a meeting of the committee."
#Dail Oireachtas Transport & Sports Cttee decided by majority vote not to invite John Delaney, FAI, before a meeting of the committee— Sean Kenny TD (@SeanKennyTD) June 10, 2015
Meanwhile, chief executive of the Scottish FA Stewart Regan has criticised the FAI for accepting money "on the back of a refereeing error".
Relations between the FAI and SFA have soured after the Irish association complained about the number of tickets made available to away fans for Scotland's 1-0 win at Celtic Park in November.
The sides meet again at Aviva Stadium on Saturday in a crucial EURO 2016 qualifier.
“It has been a challenging week for the FAI in lots of different ways,” Regan said yesterday. “I’ve always worked under the assumption if you haven’t got anything positive to say then don’t say anything - keep your mouth shut.
“The FAI have chosen to speak on a number of different fronts, recently and last November, and we’ve just got on quietly and prepared for the match and we’ll continue to do that.”
“Scotland do things professionally and we do things the right way," he said.
“It has been recalled recently that the SFA were asked to pay £75,000 into the then Fifa vice-president Jack Warner’s account for his personal use following the Scotland-Trinidad and Tobago friendly in 2004 when John McBeth was our president.
“The person who asked for the payment was sent packing with a flea in his ear and warned that if he didn’t [leave], the police would be involved.
“As far as we are concerned, we play matches the right way and qualify - hopefully - in the right way. We would not be interested in having any debate over whether or not there was compensation due on the back of a refereeing error.”