Fianna Fáil leader Micheál Martin has raised the prospect of voters having to go back to the polls due to the complications forming a new government.
Speaking on RTÉ's Six One, Mr Martin said forming a government would be difficult, adding that the view of his party was that there are "irreconcilable differences" between Fianna Fáil and Sinn Féin.
"I wouldn't rule out another general election - this is going to be so difficult," he said.
He said his party will meet with other parties to discuss the possibility of the formation of a government.
The Fianna Fáil leader said his parliamentary party colleagues "spoke frankly" at today's meeting but all were in full agreement on the decision not to enter into government with Sinn Féin.
He said he would not rule out another general election given the numbers that the electorate returned.
Mr Martin said his party will meet with "like-minded parties", and confirmed he would meet the Green Party, Labour and Fine Gael, but wondered if "a government [will] be formed that will be radical enough that it will deal with housing and health."
He said his party wanted to agree a programme for government with other parties which would "urgently deal with housing, health, climate change and the regional imbalances and making sure rural Ireland is viable in the future."
Asked about when he would talk to Fine Gael, Mr Martin said "over the weekend or next week we can talk to Fine Gael and other parties".
He added: "I don't see anything substantive happening before next week."
WATCH: Interview with Fianna Fáil leader Micheál Martin after parliamentary party agrees not to enter into talks with Sinn Féin about government formation | Read more: https://t.co/y3aOqRJ5ld https://t.co/FtA24biExu— RTÉ News (@rtenews) February 13, 2020
Sinn Féin's Louise O'Reilly said that her party does not "fear" another general election.
Speaking on RTÉ's Prime Time, Ms O'Reilly maintained the party's position that it wants a government without Fine Gael or Fianna Fáil and that Sinn Féin wants to deliver on the "mandate for change".
On government formation talks, Ms O'Reilly said Sinn Féin's message to other parties is "if you want to deliver change, we will talk", but that if that is not possible, Sinn Féin does not fear another election.
Speaking on the same programme, Fine Gael TD Hildegarde Naughton said that in terms of forming the next government, "anything is possible" down the road.
She said if required, Fine Gael would "step up to the mark" and "talk to all parties, except Sinn Féin".
However, Ms Naughton said that the onus is on Sinn Féin to carry out government formation talks.
Ms Naughton said: "They won the popular vote.
"We need to give them the time and space" to talk and not "pre-empt anything".
She said that if this fails, then the onus falls on Fianna Fáil.
Earlier, speaking on RTÉ's Drivetime, Fianna Fáil's Michael McGrath said the party has decided to appoint a team to open negotiations with the other parties.
He said it is up to Mr Martin to appoint people to this team.
Mr McGrath said his party respects the views of those who voted for Sinn Féin, but he said Fianna Fáil's parliamentary party "acknowledged the reality that the views and policies (of Sinn Féin) are so divergent, particularly on economic policies."
He said: "We got a mandate too and we have to acknowledge, in part at least that this was based on that commitment we made to voters on the doorsteps, where often people asked us directly."
The Cork TD called for clarity from Fine Gael on what he called the "contradiction between what Leo Varadkar said during the campaign, and comments made by Simon Coveney at a count centre following polling day", wherein, he said, that the likelihood of a confidence and supply type arrangement is "unlikely".
He added that his party would not be abstaining on a vote on Ms McDonald to become Taoiseach when the 33rd Dáil sits.
Fine Gael has dismissed a contention of Mr McGrath's comments that its policy on government formation has changed.
A party spokesperson said what Mr Coveney said today is the same consistent position that Mr Varadkar articulated during the election campaign.
Ms McDonald earlier told her parliamentary party meeting that it would be "quite a challenge" for Fianna Fáil to sign up to a government of change, but there is an obligation to act urgently.
She had written to the Fianna Fáil leader seeking a meeting about government formation.
She said his position not to speak to Sinn Féin is so far untenable.
Asked about the potential for a grand coalition between Fianna Fáil and Fine Gael, she said it would be the worst outcome after the General Election and a step backwards and not the government the people voted for.
Ms McDonald also said her meeting with the Social Democrats was "very good, a constructive meeting and we will talk again.
"Our strong wish is to create a government of change that is progressive and answers all the big issues that people have raised with us".
She said housing is front of centre and we are not having another five years of housing and accommodation crisis.
Ms McDonald also said there needs to be a health service that needs to be island wide in the long term, and a pension age that is set at 65.
"We are stepping our way through this process to see what this government of change might look like."