The leader of Fianna Fáil has said there is a need to move in a timely manner with coalition building, but that it will not happen before Christmas.
Micheál Martin said there are pressing matters for the new government to face, but it would be presumptuous for him to put a deadline on it given other parties are involved.
Mr Martin said: "We could have most of the work done by Christmas but we have to focus on the issues.
"I don't think we will have reached agreement by Christmas. But I’m sure that most people in the country want us to form a Government as quickly as we can."
He also said he will seek a mandate from his Fianna Fáil TDs when the parliamentary party meets tomorrow but a priority for him in any talks is that the coalition will go a full term and deliver decisive government.
Mr Martin declined to get into "details" such as a rotating Taoiseach, saying it was far too early to discuss such matters.
Following Friday's General Election Fianna Fáil has finished up with 48 TDs, Sinn Féin 39, Fine Gael 38, Independents 16, Labour 11, Social Democrats 11, PBP-Solidarity 3, Aontú 2, Independent Ireland 4, Green Party 1 and the 100% Redress Party has 1 seat.
Fine Gael and Sinn Féin will both also hold meetings of their parliamentary parties tomorrow, while the Social Democrats will hold a meeting of its parliamentary party online today.
It has written to Sinn Féin, Labour, Fianna Fáil and Fine Gael seeking meetings to discuss the next steps.
The party will hold discussions with Sinn Féin on Thursday, and will hold a meeting with the Labour party on Friday.
Watch: Former Labour Party leader denies reports of a split within the party
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Sinn Féin 'will never give up' on work for a better future
Sinn Féin leader Mary Lou McDonald has said her party "will never give up" their work for a better future, as parties hold meetings in the coming days in the hopes of forming a government.
Ms McDonald this afternoon said: "Hundreds of thousands of people voted for change. I know there is real disappointment that Fianna Fáil and Fine Gael seem dead set on picking up where they left off before the election.
"But be absolutely certain of this - we will never give up. Too much is at stake.
"The work for a better future continues right now."
Her party’s 39 TDs will meet tomorrow at Leinster House.
Labour leader Ivana Bacik is due to meet Green Party leader Roderic O'Gorman on Thursday, ahead of Friday's meeting with Social Democrats deputy leader Cian O'Callaghan.
The outcome of internal meetings between Fianna Fáil and Fine Gael could pave the way for some level of engagement on government formation before Christmas, but the intense negotiations might not start until January.
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At 86, the combined number of Fianna Fáil and Fine Gael seats is just two short of a majority.
With the total number of seats in the Dáil increased from 160 to 174, following the review by the Electoral Commission last year, any potential coalition will need at least 88 TDs.
Therefore, up to eight more TDs might be needed by Fianna Fáil and Fine Gael to ensure that a working majority can withstand any shocks along the way.
That means a new government would have to get the support of a number of Independent TDs or a smaller party.
The Dáil is due to sit for the first time on 18 December, but whether a new government could be formed within that time-frame remains to be seen.
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Meanwhile, the members of the new 34th Dáil have begun to sign the Roll of Members at Leinster House.
By signing, it enables TDs to take their seat when it convenes for the first time on 18 December.
Among the new deputies were Fianna Fáil's Tom Brabazon and Independent Barry Heneghan.
The signing of the roll takes place in the presence of the Clerk of the Dáil, Peter Finnegan.
Another deputy signing the roll was Fianna Fáil's Timmy Dooley, who was returned to the Dáil in the General Election after spending the last Oireachtas term in the Seanad.
Fine Gael's Emer Currie, the new TD in Dublin West, has also signed the Roll.
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'More cohesive progressive left bloc' after election, says SF's Ó Broin
Sinn Féin's Housing Spokesperson said the numbers are clear and it is possible for Fianna Fáil and Fine Gael to form a government with additional support.
However, Eoin Ó Broin reiterated his party's view that this is the worst possible option, and that Sinn Féin will hold talks with like-minded progressive political parties.
"Our preference is a government without either party," he said, "but Sinn Féin has always said if that isn't possible, we would talk to everyone - including Fianna Fáil."
Speaking on RTÉ’s Morning Ireland, he said it must be acknowledged that something shifted during the election.
"There is now a far larger and more cohesive progressive left, republican bloc between Sinn Féin, Labour and the Soc Dems."
"Ultimately a Fianna Fáil/Fine Gael government would be bad for those who voted for change," he said.
Mr Ó Broin acknowledged that Sinn Féin did have some disappointing take-aways from the election.
However, he said there were also positive elements and that the party turned things around during the election campaign and consolidated its position as the second largest party in the Dáil.
Independent TD Harkin interested in supporting next government
Meanwhile, Independent TD Marian Harkin for Sligo-Leitrim said she is interested in supporting the next government.
However, she said conversations should also be held between other like-minded Independents because being part of a grouping gives Independents more speaking time and a stronger position.
Speaking on Today with Claire Byrne, she said: "It certainly looks as if it will be Fianna Fáil and Fine Gael again and I would be absolutely interested in speaking to them. But first of all, there are conversations to be held with some colleagues, Independent colleagues, as to see how perhaps we can have a stronger voice."
Independent Ireland TD Michael Fitzmaurice, for Roscommon-Galway, said he has not heard from any of the three main parties.
Speaking on the same programme, he said a lot of "shadow boxing" is going on and he believes Fine Gael will play hard ball for a while in order to get "fifty-fifty on everything".
He said that the focus must be on forming a stable government.
He added that Independent Ireland is open to sitting down and holding talks with parties but that they would have to focus on policy.
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Labour had 'magnificently successful election', says Howlin
Former Labour leader Brandan Howlin has said the party "had a magnificently successful election".
Speaking on RTÉ's News at One, Mr Howlin said that there would have been "extraordinary odds" in the bookmakers’ of Labour winning 11 seats.
The former Wexford TD added that any decision to go into government with Fianna Fáil and Fine Gael will be made by the Labour parliamentary party.
"Ivana [Bacik] has said what’s going to happen - and I think it’s exactly the right tack - and that is to build a critical mass of like-minded deputies," he said.
"We have a value system; we want a different type of Irish society based on social democratic values that we've always held dear, and we want to reach out to people who have been elected to the new Dáil that share those values and then talk to anybody else, because I think that it is critically important."
Mr Howlin said that Labour is not there to make up numbers, but to "fundamentally transform Ireland", adding that his party will need to reach out to other like-minded parties.
"We're looking for how we will address the issues that are facing the people of today, most especially housing, and I honestly believe if the incoming government doesn't solve the issue of housing shortage, we will have abandoned a generation."
Additional reporting Mícheál Lehane
