Fianna Fáil leader Micheál Martin will face angry demands from TDs to allow them to serve in opposition after a disastrous General Election.
The move could fast-track Mr Martin's resignation and trigger a leadership contest.
Meanwhile, Brendan Howlin became the first senior politician to fall on his sword after he resigned from the Labour Party leadership.
Before he announced his resignation, Mr Howlin revealed Labour's six TDs would not be entering government during the next Dáil term.
The decision significantly damages Sinn Féin leader Mary Lou McDonald's plans to lead a left-wing coalition.
Meanwhile, Mr Martin is expected to tell a parliamentary party meeting today that he is opposed to entering into government with Sinn Féin.
This would leave open the door to Fianna Fáil speaking to Fine Gael about a coalition which is supported by some in Mr Martin's party.
However, senior frontbench Fianna Fáil TDs, including deputy leader Dara Calleary, Barry Cowen, Darragh O'Brien and Willie O'Dea, are opposed to entering into coalition with either Sinn Féin or Fine Gael.
Ruling out both parties significantly reduces Fianna Fáil's options, especially if Fine Gael will not support the party through a confidence and supply agreement.
Ahead of the meeting, senior party figures were suggesting Mr Martin's almost decade-long leadership will come to an end sooner rather than later. A number of TDs said even if Mr Martin manages to become Taoiseach he will not lead the party into the next general election.
A senior Fianna Fáíl TD last night downplayed a suggestion of a heave against Mr Martin, saying he will know he has to give an indication of when he will step aside.
"There shouldn't be any need for a heave and Micheál must know himself that he will have to say when he is going even if he's Taoiseach."
Another TD said: "At the end of the day he won't be leading us into the next election so he can go now or wait a year and go then."
However, others in the party were less eager for their leader to step down.
"No point in taking the captain off the pitch if you don't know who's coming on to replace him," said another TD.
Mr Martin has not spoken publicly since he left the door open to a coalition with Sinn Féin after the weekend's General Election.
Party loyalists were yesterday suggesting the Fianna Fáil leader was "buying himself some time" after the election.
Fianna Fáil TDs have been getting a backlash from their supporters over Mr Martin's failure to rule out a coalition with Mary Lou McDonald.
There is an expectation at the most senior ranks of the party that Mr Martin will rule out a coalition at today's meeting.
Meanwhile, Fine Gael is also split on the prospect of going into government with Fianna Fáil.
Yesterday, Taoiseach Leo Varadkar said he expects to lead Fine Gael into opposition but also suggested there could be a second election.
"I think the likelihood is that at the end of this process that I'll be the leader of the opposition and obviously (if) my new parliamentary party still want me to do that, I will want to do that," Mr Varadkar said.
However, some Fine Gael ministers believe they should approach Fianna Fáil about forming a coalition.
But the Taoiseach's view is supported by other senior figures in the party.
Meanwhile, Sinn Féin's negotiating team opened talks with the smaller parties yesterday as Ms McDonald sought to form a government.
The Green Party and People Before Profit held talks with the Sinn Féin team, which is led by Pearse Doherty.
Speaking after the meetings, Sinn Féin negotiator Louise O'Reilly said her party's objectives were to build homes, cut rents and freeze them, reduce the pension age to 65, give workers and families a break and advance Irish unity.
The Social Democrats are expected to hold talks with Sinn Féin today.
Ahead of the meeting, the party's co-leader Róisín Shortall said it was unlikely the next government would be a left-wing coalition supported from the opposition benches by either Fine Gael or Fianna Fáil.
"Confidence and supply didn't work very well in the last Dáil and I think it is unlikely to be a composition in this Dáil," she said.
She added her party planned to also hold talks with Fianna Fáil, the Green Party and the Labour Party.
In an early sign of the difficulties in forming a left-wing government, the party called Solidarity issued a statement yesterday clarifying that it did not attend a meeting with Sinn Féin.
"The meeting today is just a meeting of the People Before Profit component of Solidarity - People Before Profit, with Sinn Féin, arranged between the two parties themselves," it said.
Solidarity currently only has one TD, Mick Barry.
Einstein famously said that the definition of insanity was doing the same thing over and over and expecting a different result. And it appears that the Irish people, after decades of doing the same thing, have finally decided to take a risk by doing something very different and voting for Sinn Féin in order, one presumes, to get a different result.