Sinn Fein president Gerry Adams has called on Fine Gael and Fianna Fáil to form a government.
With the Dail returning on Thursday in deadlock over a future coalition, Taoiseach Enda Kenny is expected to remain as "caretaker" while pressure ratchets up on him to do a deal on an alliance with his political arch rivals.
"These two parties have the same political outlook, the same policies, and the numbers to govern. These two parties have the votes to elect a Taoiseach tomorrow," Mr Adams said.
"Those on hospital trolleys and those sleeping in doorways should not be left waiting while Fianna Fail and Fine Gael jockey for position."
Three nominees are to be put forward tomorrow for the role of Taoiseach: Mr Kenny, who has held the position since 2011, Michael Martin, who has led a resurgent Fianna Fail, and Mr Adams - with no-one likely to get the necessary support.
Once the outcome is known, Mr Kenny will visit President Michael D Higgins in Aras an Uachtarain where he will either resign or ask for the Dail parliament to be dissolved, sparking a second election in just over a month.
Most politicians and commentators accept there is no appetite among voters to rehash the lacklustre campaign and the subsequently divisive poll.
Ireland's Constitution provides for the outgoing Taoiseach and Government ministers to remain in office until the members of the Dail agree on their replacements.
There is no time limit on how long caretaker roles can be held but one anomaly is that all members of the cabinet must have retained their seats in an election, meaning several positions at the table need to be filled.
On the back of a massive voter schism, a previously unthinkable alliance between arch rivals Fine Gael and Fianna Fail appears to be the only viable scenario.
Mr Kenny and a handful of top negotiators have met smaller parties and Independents over the past few days but a stable government is unlikely with them due to the patchwork make-up of the parliament.
He has offered limited insight into potential coalitions other than to commit to work with rivals.
Mr Kenny also vowed that any administration his party leads would deal with issues "highlighted and prioritised" by voters during the election campaign - unprecedented homelessness, the debacle of water charges and the Irish Water utility and the overcrowding crisis in the health service.
And while there is little policy difference between centre-right parties, huge swathes of their memberships retain bitter animosities borne out of Ireland's civil war more than 90 years ago.
There is also concern in Fine Gael that it would face further voter backlash if it implements difficult reforms and policies, even with Fianna Fail support.
Both parties are also wary of handing the official opposition to Sinn Fein, the third largest party, which has refused to prop up either of them.
Mr Adams called for the Dail to get back to business rather than be adjourned for lengthy negotiations on a new coalition, alliance or minority government.
Mr Martin called for parliamentary reforms before any deal is tabled on a coalition, alliance or minority government.
Labour, the outgoing junior coalition partner which was near wiped out after holding just seven seats, is to support Mr Kenny's nomination as Taoiseach.
Elsewhere, the 158 Dail seats are filled by the most diverse collection of politicians in the Republic's 95 year history.
Among those who may yet hold the balance of power are Independents, who took 17 seats, and the Independent Alliance, which has six seats and describes itself as like-minded, radical but responsible politicians.
There may also be a small hand to play for the Social Democrats who have three high-profile TDs - Stephen Donnelly, Roisin Shortall and Catherine Murphy - or the Greens who returned to the Dail with two seats.
The sitting will also see the election of a new Ceann Comhairle, a chairman or speaker.
The vote will be in secret and the five candidates are Independent Maureen O'Sullivan, Fianna Fail's Sean O Fearghail, Fine Gael's Bernard Durkan and Andrew Doyle, and Sinn Fein's Caoimhghin O Caolain.
