Sinn Fein leader Mary Lou McDonald has criticised what she described as the "old boys' club of Fianna Fail and Fine Gael" after they have both refused to enter government talks with her party.
Fianna Fail agreed not to go into coalition with Sinn Fein following a four-hour meeting of its parliamentary party on Thursday.
The decision is likely to stop Sinn Fein entering power, as their leader Mary Lou McDonald said forming a coalition without them or Fine Gael would be "very, very tricky".
Ms McDonald said: "Any suggestion that Fianna Fail and Fine Gael together suggest change is farcical - transparently farcical.
"I think it is actually quite disgraceful that the old boys' club of Fianna Fail and Fine Gael believe that they can set aside the democratic mandate of Sinn Fein and the electorate."
She said it would be “unthinkable” for Fianna Fáil and Fine Gael to be in power for another five years.
McDonald said she accepts her number one option for a purely left-wing government is “off the table”.
But she insisted there are still strong numbers for a “very strong government of change” to avoid the traditional scenario of a “big monster party with a small party tacked on”.
“The numbers now are such that I accept that we can’t do a government now,” she said, at a lunchtime rally by striking community employment supervisors who are demanding a pension. My number one option I guess is off the table.
“What’s not off the table is the real prospect, possibility, that we can deliver a government that’s not like the governments of the past, where you had a big monster party and some small party tacked on.
“And a government that has Sinn Féin in it and a government that most importantly, above all of that, a government that actually is responsive to working people. “
She said other people had a very good day out at the polls, including the Social Democrats, the Greens, and “progressive-minded” independents.
“So all of the ingredients are there to create this government for change,” she said. “We can do it alright.”
She said the worst possible outcome would be five more years of Fianna Fáil and Fine Gael.
“That to me is just unthinkable,” she said.
She said that was not what people voted for and people were crystal clear that they wanted change.
“The maths are difficult in the Dáil, that’s for sure, but there is still a critical mass of TDs that are elected, like the Sinn Féin TDs, on an agenda of change and delivering on housing, on cutting rents and freezing them, on getting the pension age back to 65. So we can still craft a government that can do all of these things.
“What’s happened now, it seems to me is that Fianna Fáil and Fine Gael want it all their own way again and they want to push out the alternative, they want to exclude us.
“And I for the life of me, I cannot fathom how anybody would think it a good idea to have five more years of Fianna Fáil and Fine Gael. That is not the view of the electorate or the general public.
“This isn’t about me personally even, or my chances, this is about actually having a government that can deliver properly for people. That’s the issue here.
“Fianna Fáil and Fine Gael have had it their own way for almost a century and the idea of more of the same, I think would actually cause great anger amongst the people and the electorate. “
Meanwhile, Eoin Ó Broin said that even if all of the newly elected "progressive left" TDs were on board with a Sinn Féin led left-wing government, there still would not be enough to make up a cabinet.
Asked how Sinn Féin can form a government without either Fianna Fáil or Fine Gael, he replied: “You can’t.”
"If you listen to what Mary Lou said in the last two days, it's very clear that the only stable government is going to involve two of the larger parties and that's why Mary Lou wrote to Mícheál Martin the evening before yesterday to seek initial discussion with him," he said.
"Our preference was to have a government of the left without Fianna Fáil or Fine Gael, but from Tuesday morning, once you could see the shape of the Oireachtas, the broad progressive left has about 66 TDs and that's never going to be enough for a government.
"In real terms that means we need to talk to one of the larger parties to see if either one of those wants to participate in that government."
Fine Gael has been consistent since the election was called that they would not talk to Sinn Féin about forming a government.
Fianna Fáil was also adamant that they would not form a government with Sinn Fein but as the campaign progressed, Mícheál Martin's stance began to soften.
A number of Fianna Fáil sources also said they would be open to enter talks with Sinn Féin, however, after first Fianna Fáil's parliamentary meeting after the election, the party's leader hardened his stance again.
Mr Martin said that his party were united "in relation to Sinn Féin" in that "people felt the economic platform that Sinn Féin put forward was irreconcilable with Fianna Fáil, particularly on the enterprise agenda and sustainability part."
Mr Ó Broin agreed that "there are huge policy differences" between the parties but said he would still be willing to work with Fianna Fáil and wouldn't rule out Mícheál Martin changing his mind again.
He did, however, criticise the Cork TD for the inconsistency in his stance on a coalition, and the threat it poses of another election.
"The only responsible thing to do for any party is to sit down and talk to all to talk resolve those issues and deliver a government to the people that delivers real change on the issues of importance to the electorate," he said.
"We are waiting for a formal response from Mícheál Martin. Mícheál Martin over the last number of days has said various things and there isn't a clarity.
"This is a man that said during the general election that he wouldn't talk to Fine Gael either, yet yesterday he suggested that he's going to put Leo Varadkar back into office and give us the same kind of government as the last four years.
"If last week Mícheál Martin was saying that he isn't going to go into talks with Fine Gael and this week he says he is, then very clearly Mícheál Martin changes his mind on these matters.
"Political parties who are saying they won't talk to others, who are threatening elections, don't just smack of arrogance, but I actually think it's reckless. The 4,000 children in emergency accommodation can't wait for this government to be formed. It needs to happen as quickly as possible."
The Dublin Mid-West TD said that his party are talking to the Green Party, the Social Democrats, and "other progressive TDs".
He said that Mary Lou McDonald will decide whether to write to Fine Gael in the coming days, a party who, despite consistent resistance from whom, he would still consider forming government with as strongly as he would Fianna Fáil.