Fianna Fáil and Fine Gael have said they had "positive and constructive" initial discussions as both parties continue to hold government formation talks.
In a joint statement, they said "both parties will continue to engage on issues over the coming days".
The Social Democrats, meanwhile, are likely to make a decision by the end of this week on what next steps to take, if any, in government formation talks.
The party's deputy leader Cian O'Callaghan confirmed the likely schedule this evening, after the Social Democrats negotiation team met with Fianna Fáil and Fine Gael at Government Buildings.
Mr O'Callaghan and party colleagues Gary Gannon and Jennifer Whitmore met with Fianna Fáil leader Micheal Martin and deputy leader Jack Chambers, as well as Fine Gael leader Simon Harris and his colleagues Helen McEntee and Paschal Donohoe.
Speaking to reporters after the separate meetings ended, Mr O’Callaghan said the Social Democrats' parliamentary party will be "meeting tomorrow and reporting back on those meetings".
Asked if a decision on whether to continue with talks or not will be made, he added:
"We're going to update the parliamentary party tomorrow on the talks we've had, and at that point we'll decide what the next steps are.
"There could be further engagement, but we do need to discuss it at our parliamentary party," before adding when asked if he believes a decision will be made by the end of the week:
"Yes I do, yeah."
The deputy leader of the Social Democrats was also asked if his party or he personally would be in favour of rejecting going into government and instead forming an opposition left alliance, he said the Social Democrats are currently discussing government formation and that they have not considered the alternative at this stage.
Members of the Regional Group of Independents, meanwhile, confirmed that they were meeting, as talks continue to take place between Fianna Fáil, Fine Gael and smaller parties.
Speaking to RTÉ News on his way into Leinster House, newly elected Independent TD for Dublin Bay North Barry Heneghan said he is "still just chatting to everyone, getting calls from every side".
Deputy Heneghan, who is being advised by former TD and government minister Finian McGrath, said his priorities include "a new A&E" for his constituency.
The Regional Group confirmed that it has agreed to nominate Wexford TD Verona Murphy for the position of Ceann Comhairle.
Independent TD Carol Nolan has also now joined the group.
Fianna Fáil leader Micheál Martin earlier said he was hopeful that substantive work could be completed before Christmas.
Mr Martin said he knew people were anxious to form a government after the General Election and that when the Dáil resumes in the New Year it would be ready for the election of a Taoiseach.
Asked if that would be him, Mr Martin said he was going to go through discussions and he respected all parties involved in those discussions.
Third leg of next government beginning to look clearer
The Regional Independent Group is now firmly on course to become the third leg of the next government.
The group has nine TDs: Michael Lowry, Noel Grealish, Kevin 'Boxer' Moran, Seán Canney, Marian Harkin, Verona Murphy, Carol Nolan, Barry Heneghan and Gillian Toole.
It will appoint a negotiating team next week for government formation talks with Fianna Fáil and Fine Gael.
Before that Michael Lowry, the group's lead spokesperson, will sit down with the deputy leaders of Fianna Fáil and Fine Gael, Jack Chambers and Helen McEntee, to put in place the parameters for these talks.
This follows on from a conference call between Michael Lowry, Simon Harris and Micheál Martin in recent days.
One thing is already clear, the group wants to be a fully integrated part of any new government.
Read Micheál Lehane's full analysis
Murphy accuses Bacik of contradiction in coalition position
People Before Profit-Solidarity TD Paul Murphy has said Labour leader Ivana Bacik is contradicting herself by telling the public she is interested in entering a government while telling party members she will not.
The Irish Independent reported this morning that Ms Bacik was recorded telling her party's ruling body that she has no intention of going in to government with Fine Gael or Fianna Fáil and would only speak to them as a courtesy.
The newspaper reported that notes of a Labour central council meeting last Saturday show Deputy Bacik told the meeting she did not believe the party secured a strong enough mandate to enter coalition.
"We will meet the big three as a courtesy, but with no intention of going into government," the Labour leader is quoted as saying, in the notes compiled by a party representative.
Deputy Murphy said "we have Ivana Bacik saying two things", saying the Labour leader is "telling the media she is interested" and to her party "not really. Which one wins out, I don't know".
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