Minister for Communications, Climate Action and Environment Richard Bruton has said taxpayer's money should not be paid into a "black hole" in order to fix the FAI's problems.
Speaking this afternoon in Dublin, Mr Bruton also insisted that nobody wanted to see the association go into examinership or liquidation, which FAI Executive Lead Paul Cooke yesterday said was a possibility.
"I have no doubt that, as the Minister himself (Shane Ross) has said, he does not want to see examinership or liquidation as the way forward for the FAI," said Minister Bruton.
He added: "Hopefully there is a space now emerging where the Minister can work with the organisation to develop the strategy that, I think, we all want to see.
"But, equally I don't think anyone wants to see taxpayer’s money going into a black hole.
"This has to be run in a way that we can stand over it and that any money which is given from the taxpayer is accounted for in a transparent way".
Earlier, Minister for Health Simon Harris said the Government wanted to support Irish football, but would not be writing a blank cheque for an organisation that it does not have confidence in.
He said the FAI has been run like a "fiefdom" and a personal club, which he said "stinks" and portrays an arrogance he thought had left with the Celtic Tiger.
He welcomed yesterday's apology, but said the FAI did not elaborate on what the apology was for.
The association said it was sorry yesterday evening to the "hundreds of thousands involved with Irish football at all levels of the game".
The FAI has current liabilities of €62m and is facing the prospect of examinership or liquidation if it cannot find at least €18m in refinancing.
The Chair of the Oireachtas Committee on Transport, Tourism and Sport said football, particularly the grassroots, cannot be allowed to fail and the Government will play its part, but regime change is needed at the FAI.
Fine Gael TD Fergus O'Dowd said there is money for sport, but that cannot mean the writing of a blank cheque.
Stuart Gilhooley, a solicitor for the the Professional Footballers' Association of Ireland, said it was clear that the FAI was very close to examinership.
Mr Gilhooley, who attended yesterday's FAI AGM, said the most important thing to focus on is the future and how to get the FAI "out of the massive hole that it's in".
He said it was inevitable that someone will have to cough up money in some shape or form and it was possible that the Government and UEFA could do some sort of deal.
He added that this situation was unprecedented and a serious indictment of those involved.
A former chairman of the FAI National League said it was hard to believe the FAI could be allowed "go to the wall".
Brendan Dillon, who is also a former officer of the FAI, said that tens of thousands of people voluntarily give their time to the FAI and asked if they should be punished for the sins of the few.
He welcomed the apology given by the board of the FAI yesterday, but said there were serious and stark financial questions to be asked of the association.
Mr Dillon said it appeared that a meeting has been arranged with UEFA who, he said, "would not want one of its members going to the wall".
Meanwhile, Minister for Sport Shane Ross said it would be "welcome" if Republic of Ireland manager Mick McCarthy took a pay cut given the financial difficulties affecting the FAI.
Additional reporting: Maggie Doyle