Minister for Sport Shane Ross has said the Football Association of Ireland was looking for financial support of "over €10 million" from the Government.
Speaking following a meeting with FAI executives at Government Buildings, Mr Ross said the association told him it had a serious financial crisis and was seeking "unthinkable amounts", but there would be no bailout for it.
The meeting happened at FAI's request, to discuss the financial situation, which was revealed ten days ago in the governing body's accounts.
"We were not in the business of bailing out the FAI in any circumstances," Mr Ross said.
He added: "There was no way the Government could possibly think of giving it to them. There have been serious recommendations made to us not to fund the FAI at present".
The minister also said he will be seeking a meeting with UEFA "because they are also an interested party and see if there's any solution that doesn't involve a bailout by the Irish Government.
"We'll seek that meeting tomorrow morning.
"If UEFA and ourselves can come across with support for football in Ireland, we'll do what we can to support it," he said.
Vice President and Executive Lead of the FAI Paul Cooke described the talks as a "good, constructive meeting".
He said that the FAI will not be meeting with the minister in the short term but "there will be meetings".
Earlier, the minister described the FAI as "a basket case" and said he does not care what happens to the association, but does care about the future of Irish football.
However, speaking to RTÉ News during a visit to Port of Waterford, he described the situation facing the FAI as "really very grave".
Asked if he expects the FAI to exist as it does in a year's time, Mr Ross said: "The FAI is in a really very grave situation at the moment. I can't over-estimate how bad things are and they've brought it upon themselves.
"What I'm interested in is the future of Irish football, I'm not interested in the future of the FAI. It’s a basket case, it's in a really grave place.
"What I want to see is that the grassroots of Irish football are looked after, that those things that you see here in Waterford and that we see elsewhere in Ireland that happen on Saturdays and Sundays and every day of the week, with the volunteers and grassroots, are doing fantastic work.
"The FAI has, unfortunately, done some things which are not easy to reconcile with that magnificent work which is done at the grassroots. I don't care about the FAI. I care about Irish football."
Additional reporting Conor Kane