Fianna Fáil has agreed to negotiate with Fine Gael to facilitate a Fine Gael minority government.
At a meeting this evening, many members said Fine Gael would have to secure 58 votes to make it viable.
Several members also said they were against a written document emerging from the negotiations and preferred a situation where the party would support the minority government on a case by case basis.
The negotiating team was given discretion in the talks but will report back to the parliamentary party.
Members also said they wanted party policies to be implemented.
The Dáil adjourned tonight until Wednesday 20 April, after failing to elect a taoiseach for the third time.
Enda Kenny secured 52 votes for and 77 against.
Micheál Martin secured 43 votes in favour to 91 against, but he did not get any votes from the independents.
Speaking on RTÉ's Six One Fianna Fáil's Charlie McConalogue said party leader Micheál Martin will not put himself forward for taoiseach again.
Mr McConalogue said Fianna Fáil believes that things need to move on and there needs to be a government.
Fianna Fáil’s Charlie McConalogue confirms Micheál Martin will not be putting himself forward again as taoiseach https://t.co/sSD9vCSRUx
— RTÉ News (@rtenews) April 14, 2016
Leo Varadkar, who is on Fine Gael's negotiating team, said that in return for the facilitation of a Fine Gael led minority government, it would accept some of Fianna Fáil's policies.
Speaking on the same programme, he said that Fine Gael will need to get independents or smaller parties on board to get to 60 to ensure a viable government.
Leo Varadkar says FG needs support of Independents and possibly other parties to be a viable and stable govt https://t.co/IJckhynZ1T
— RTÉ News (@rtenews) April 14, 2016
Following today's failure to vote in a taoiseach, Mr Kenny invited Mr Martin to authorise his party's negotiating team to resume talks with Fine Gael.
Mr Martin told the Dáil it was time to move on and he said they would continue to participate in talks on a minority government.
Last night Fianna Fáil upped the ante, saying it was the last opportunity for independents to support Mr Martin as taoiseach and in effect to support a Fianna Fáil minority led government.
However the move had no impact on the outcome of the vote as 14 of the 15 independents said they would be abstaining on the votes for both leaders in the absence of an agreement between the two parties.
Independent TD Michael Lowry and the 50 Fine Gael TDs voted again for Enda Kenny.
In a surprise move, Independent Katherine Zappone, who had earlier indicated she was abstaining, voted in favour of Mr Kenny.
As it happened: Third attempt to choose taoiseach
Speaking in the Dáil Mr Martin said it was time to abandon the maneuvering and inflexibility.
FG-FF argument a 'long drawn out domestic dispute' - McDonald
Sinn Féin deputy leader Mary Lou McDonald described the antics between Fine Gael and Fianna Fáil as Tweedledum and Tweedledee who have been engaged in a long, drawn out domestic dispute.
She said she does not want to see either of the two as taoiseach, but said the numbers are the numbers – 52 to 43.
Ms McDonald said that Mr Kenny and Mr Martin need to make up their minds but that they should not insult people by prolonging "this farce".
Before the vote Labour leader Joan Burton said the current process had left a lot of people scratching their heads.
She was heckled as she said Sinn Féin TDs were sitting on their hands and shouting from the sidelines, with no intention of putting their shoulder to the wheel.
Independent Alliance TD Shane Ross said the previous weeks of discussions with both parties had been progressive and they had got used to being love bombed by both sides. The problem was that they would not love bomb each other.
He said that Civil War politics immediately emerged when Fianna Fáil and Fine Gael began to negotiate.
Mr Ross said it was time to end the playacting and take immediate steps to form a government.
The news that 14 Independents would abstain in today's vote came this morning.
The 14 TDs, including Independent Alliance, the Rural Alliance and Michael and Danny Healy-Rae, also decided to withdraw from talks with Fianna Fáil and Fine Gael on the possible formation of a new government, until they come to an agreement with each other.
In a statement they said: "We want the parties to agree on a three-budget programme for a minority government and we are withdrawing from the talks process until that can be agreed," the deputies said in a statement.
"We have also made it clear that if the main parties want a facilitator to assist with their talks, that will be made available to them."
