Fine Gael Ministers offered Fianna Fail a six to nine month suspension of water charges at a meeting on Monday.
The negotiating team, which consists of five senior Ministers, tabled a temporary suspension of the levies while an independent commission carries out its work.
Fianna Fáil strongly rejected the proposal made and insisted it could not settle for a short term suspension.
The talks, which lasted three hours and will resume tomorrow, focused primarily on Irish Water and “frank exchanges” were made between the two parties. Relations between the two parties are understood to be at their lowest since negotiations began.
Fine Gael insisted Fianna Fáil made no counter proposal to their offer and requested one be made at a meeting scheduled for tomorrow morning.
However Fianna Fáil is insisting it suggested a series of alternatives to the offer. Fianna Fail offered Fine Gael a three year agreement in return for the suspension of water charges for three years.
At a meeting on Monday the party’s negotiating team said Fianna Fail was willing to give three budgets in return for the levies being paused.
The length of any agreement would be linked to how long the charges would be suspended for.
Minister for Agriculture Simon Coveney said Fianna Fáil was not in a position to make a counter offer on Monday but he hoped they would be able to at a meeting on Tuesday morning.
He said: “A lot has been agreed. There was 30 items up for discussion, the vast majority of them have been agreed in terms of accommodation or compromise.
“There are a relatively small number of issues that are not resolved one of them is water. The Taoiseach tried to move it on at the weekend and we would like a response to that proposal from Fianna Fail and I hope we will get that in the morning.”
The Fine Gael parliamentary party meeting is expected to meet on Tuesday to discuss the latest negotiations.
Mr Coveney insisted the party would not go beyond what their parliamentary party finds reasonable or acceptable.
He said: “Both parties will have to make compromises to accommodate each other. Fine Gael has done that over the past number of days and we hope we will get a response from Fianna Fail in the morning.”
A Fine Gael source said the next 48 hours would be crucial in the talks and claimed significant compromises would need to be made or an election would have to be called.
However speaking after the meeting Fianna Fáil TD Barry Cowen said his party was wedged to its commitment of suspension for five years.
He said: “We’re seeking to facilitate Fine Gael in government, we’re not seeking to go in with them.
“We’re seeking for them to adhere to certain guidelines and principles that we want to reaffirm. It’s our job to do that and give some value to the votes people gave us. It’s still a 50/50 chance.”
Speaking on her way into the meeting acting Minister for Justice Frances Fitzgerald said she was confident a deal could be reached but said there were difficult issues dividing the parties, in particular, Irish Water.
Her colleague on the Fine Gael negotiating team, acting Minister for Finance Michael Noonan, said progress had been made and the sides would “give it as long as it takes”.
Speaking in Co Cork on Monday, before the resumption of the talks, Fianna Fáil leader Micheál Martin said he believed the chances of success in the discussions were “very much 50/50” but must be concluded within days if they were to be successful.
Mr Martin said he was aware nobody wanted another general election but it was important that the issue of Irish Water was addressed in advance of the formation of any government.
The acting Taoiseach Enda Kenny met Fine Gael Ministers on Monday morning to discuss a proposed compromise on Irish Water.
Mr Kenny is understood to have sought guidance from his team on what the next step should be as Fine Gael and Fianna Fáil attempt to end the political deadlock to form a government.
Mr Kenny is proposing a temporary suspension of the charges while an independent commission explores a new charging regime in return for a firm commitment they would re-introduced and a written three year agreement.
However, Mr Martin said he could not promise the return of water charges during the 32nd Dáil due to the huge public opposition.
Instead, Fianna Fáil wants the commission's conclusions to be presented to an Oireachtas committee which will debate the findings and decide the way forward.
Speaking on Monday, Roscommon-Galway Independent TD Denis Naughten said he wanted Fine Gael and Fianna Fáil to strike a deal that would result in a government with the potential to operate for five years and not just five weeks.
Carlow-Kilkenny Fianna Fáil TD John McGuinness said lengthy talks were the price of good governance.
"It's important to get it right," he told RTÉ's Seán O'Rourke Show.
He agreed there were other issues that should be focused on, but said it was important that the issue was dealt with properly. “If it’s not sorted we will have to be extremely careful.”
On the suggestion of setting up a commission to decide on the future of Irish Water, he said
Minister for Health Leo Varadkar is to meet members of the Independent Alliance on Monday in a bid to secure their support for a Fine Gael-minority government.
He has offered Waterford TD John Halligan a clinical review of the 24-hour care at the regional hospital and an extension of hours.
Fine Gael is hopeful it can win the support of the six members of the Alliance, including Mr Naughton, as well as Michael Harty, Michael Healy-Rae and Maureen O'Sullivan.
Noel Grealish, of the rural alliance, is also said to be strongly considering supporting the party.