The FAI is denying there was any large pay-off for Giovanni Trapattoni.
The association announced earlier that it was parting company with the Italian and his assistant Marco Tardelli by "mutual consent".
In a statement, Trap has thanked those who supported him over the past five and a half years, saying he leaves the country with "emotion".
The FAI said its board will meet in due course to discuss who will succeed the 74-year-old as Ireland manager.
Chief Executive John Delaney said talk that Trapattoni received a handsome compensation payment is not true.
He said: "There was talk in the media about a contract up until June of next year, that particular part wasn't true.
"Everything is confidential. It was amicable, in terms of the monetary monies, it wouldn't be something substantial, I can say that at least."