FAI chief executive John Delaney said he believed they had a case against FIFA after French striker Thierry Henry's handball in the 2009 match.
Delaney said a "legitimate agreement" was made to drop the claim after Ireland missed qualifying for the 2010 World Cup.
His claim comes after a turbulent nine days for football's world governing body amid claims of "systemic and deep-rooted" corruption.
FIFA is facing two criminal investigations by both the US for widespread bribery allegations and Swiss authorities over the 2018 and 2022 World Cup bidding processes.
The Republic of Ireland were 16 minutes away from a penalty shootout for a place in the 2010 finals in South Africa before Henry handled in the build-up to William Gallas's extra-time equalizer in the second leg of the play-off in Paris.
The 2-1 aggregate win sent France to the 2010 tournament, where they were knocked out of the group stages amid a player strike.
A FIFA spokesman confirmed the payment, but said it was a $5m loan and was granted for "the construction of a stadium in Ireland".
The spokesman added that the FAI was to repay the money if Ireland qualified for the 2014 World Cup in Brazil, which they failed to do. FIFA wrote off the loan on December 31, 2014.
