Coalition and opposition leaders are due to meet at 8.30am at Government Buildings to try and overcome the issues which led to the Dáil being suspended.
It is understood that a text has been drafted which will be discussed, with the assistance of the Ceann Comhairle Verona Murphy.
If agreement can be reached, it will clear the way for the Fianna Fáil leader Micheal Martin to be nominated as Taoiseach shortly after the Dáil reconvenes at 10.30am.
Mr Martin said the failure to elect a taoiseach and was "the subversion of the Irish constitution".
He said it was a "premeditated" and "coordinated" effort by the opposition, and particularly by Sinn Féin.
Mr Martin said there were numerous attempts to facilitate an agreement, but he said it became clear to "us" that there was "no intention ever to reach agreement on the nomination of a taoiseach or a government today".
"This is the first time, I think, in over 100 years that the Dáil has failed to elect a Government to fulfil its constitutional obligation..."
Mr Martin was due to be nominated for the position, but that process was delayed throughout the day after a row erupted over Dáil speaking time for Independent TDs who are supporting the incoming government.
This move was roundly rejected by the opposition, as was the proposed suspension of the Dáil until 5 February following the conclusion of today's business.
When the Dáil did resume the Ceann Comhairle attempted to begin the nomination process.
The opposition objected to that move and Ms Murphy adjourned the Dáil amid chaotic scenes.
The Dáil will reconvene at 10.30am in the morning.
'Utterly farcical'
Fine Gael leader Simon Harris described today's events in the Dáil as "utterly farcical".
"There was a clear majority of the people's representatives read, willing to elect Micheál Martin as Taoiseach and to elect the next Government and get on with the people's work and there is much work to done."
Instead, he said there was an effort to "obstruct" and to "disrespect" the Ceann Comhairle.
"And ultimately preventing a majority of the democratically elected representatives of Dáil Eireann to discharge their duty and their mandate to elect a Taoiseach.
"Today was unprecedented, unseemly and tomorrow Dáil Éireann must reconvene and must elect a Taoiseach.
He said when people turned on their televisions today they saw "stunt politics" and that is "utterly disgraceful".
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A day of Dáil suspensions
Before midday the Dáil had already been suspended twice after opposition TDs repeatedly clashed with new Ceann Comhairle Verona Murphy.
All opposition parties questioned the decision to allow Independent TDs who are supporting the Government to have opposition speaking time on a temporary basis through a technical group.
Concerns were also raised over the Dáil not being due to sit again until 5 February.
After the Dáil was suspended a second time, a number of meetings took place between party whips, who put forward a proposal to allow Sinn Féin more speaking time and to vote tonight on whether the Dáil will return early.
But that was not agreed when the Dáil sat again after 4pm, with the Ceann Comhairle saying the Dáil was being suspended until tomorrow morning.
Separately, in what is a potentially key development, it emerged that Aontú's two TDs will leave the Regional Independents technical group at the centre of the stand-off and join with Independent Ireland and Independent TD Paul Gogarty.
The move puts pressure on Independents Mattie McGrath and Carol Nolan, who the Regional Independents need in order to meet the bare minimum number of TDs needed to form a technical group in opposition.
Read more:
Sinn Féin to oppose plans to adjourn Dáil for two weeks
All eyes on who is getting what brief in the new Cabinet
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