Fianna Fáil has confirmed it will not propose party leader Micheál Martin as taoiseach or pursue a minority government.
Donegal TD Charlie McConalogue told RTÉ Six One the party leader would not be putting his name forward for taoiseach next week.
The Fianna Fáil parliamentary party is meeting Thursday evening to discuss its next move.
Acting Taoiseach Enda Kenny has invited Mr Martin to further talks after both men failed for the third time to be elected taoiseach in the Dáil Thursday afternoon.
The motion to elect Mr Kenny as taoiseach was defeated by 77 votes to 52.
He had one extra vote this time around with the support of Independent Katherine Zappone.
Independent Michael Lowry, who supported Mr Kenny last time also backed him in this vote.
The motion to elect Mr Martin as taoiseach was defeated by 91 votes to 43.
Mr Martin only received the support of his parliamentary party.
It was the third time the Dáil met in an attempt to elect a taoiseach.
Following the votes, Mr Kenny told the Dáil he would ask Mr Martin to meet again to discuss the resumption of talks to form a government.
Mr Kenny said he fully respected the rights of Independents to abstain from votes on the formation of a government until its composition and policy-programme were agreed.
He recalled he had proposed a full partnership government, involving Fine Gael, Fianna Fáil and Independents, believing it was the best option to provide a stable and lasting government.
“We are now nearly seven weeks on from the general election and it is time really to focus, with a sense of urgency, on what it is we have to do here,’’ he added.
Mr Martin said that in the wake of the votes “it is time to move on”.
His party was “prepared to continue in discussions with Fine Gael about the operation of a minority government”.
But the Fianna Fáil leader said Fine Gael would have to do “what they have so far refused to do, which is to detail who it expects to participate in such a government and state that government’s programme”.
Mr Martin also said Fine Gael had to “move away from the highly-controlling attitude which defined the last five years. We await a demonstration that it knows how to respect the interests of others”.
Sinn Féin deputy leader Mary Lou McDonald rounded on Mr Martin and said that if he wanted to get away from control and spin “why are you so controlling and gripped with” beating Mr Kenny on the votes, and then saying it was not about the numbers.
The Dublin Central TD said it was all about the numbers and it was about the “two big beasts throwing the rattle out of the pram because you don’t have it all your own way anymore”.
Independent TD Katherine Zappone, who voted for Mr Kenny as taoiseach said she agreed with other Independents to “hasten the formation of a government and give stability”.
The Dublin South-West TD said she hoped Fianna Fáil and Fine Gael “can give support to each other”.
Everyone was calling for a government to be formed, that was why she had made her decision.
Earlier, the Independent Alliance and the rural Independents said they were pulling out of discussions with Fianna Fáil and Fine Gael.
Shane Ross of the Independent Alliance said this was to allow the two parties come to some conclusion.
In a jibe at Mr Martin, Mr Ross said “we do not like receiving ultimatums from any party that we should vote one way or get off the pitch”.
And new Green Party TD Catherine Martin warned that the “abysmal lack of leadership might yet require the action of the President” to resolve it.
Mr Ross said the Independents had been “love bombed” by the two largest parties but they were not love-bombing each other.
In her maiden speech in the Dáil, the Green Party TD said the vast majority of people did not care in which order the names of Fine Gael and Fianna Fáil came.
But she said “now is the time to show real leadership, stop hiding behind what the parties claim is in the national interests, when it is clearly not”.
Labour Party leader Joan Burton said her party would not be supporting any of the nominees for taoiseach.
Ms Burton said in the light of the failure of Fine Gael and Fianna Fáil to produce a programme for government to address issues in a serious way, Labour was not in a position to support any candidate for taoiseach before the House.
“We are now at a crossroads,’’ she added. “I said last week the civil war is over and we need responsibility parties to put our country first.’’
Ms Burton said those who had a mandate to form a government should do so.