Fianna Fáil leader Micheál Martin has said that nothing should be taken for granted about the outcome of negotiations about the formation of a new government.
His party secured the largest number of seats in the 34th Dáil, with 48. Sinn Féin has 39 seats while Fine Gael has 38.
Both Mr Martin and Fine Gael leader Simon Harris are in Edinburgh for the British Irish Council meeting.
Speaking to RTÉ News, they both indicated that they are likely to have informal discussions about government formation on the margins of the event.
"It's a very busy night, tonight and tomorrow, so opportunities will be limited enough, obviously there are informal opportunities for us to touch base," said Mr Martin.
"But we will have time next week as well to get into more substantive discussions and we will be getting into substantive discussions early next week in Dublin, but there are always informal opportunities here to touch base."
Asked if it could be taken for granted that Mr Harris will be once again be Taoiseach as part of a rotational agreement, he replied that there is not anything to be taken for granted.
"I think we have to respect every party and I think until we get into substantive talks, I'm not presuming anything," he said.
"We have sounded out a number of independents this week, I'm due to meet with the other political parties next week, so I think, in earnest, there will be meetings next week.
"I take nothing for granted, and until agreements have been reached and completed, nobody can take anything for granted."
The Fianna Fáil leader said discussions would take place in an atmosphere of "mutual respect all round and fairness" as the key approach.
Meanwhile, Mr Harris has said he believes that Fine Gael has been given a mandate to play a pivotal and leadership role in government.
He agreed that nothing can be taken for granted at this stage, but added that he does have fundamental principles which will be brought to discussions.
"Principles around mutual respect, around parity and around policy," he said.
"But of course, I look forward to having a chat with Micheal, maybe even an informal beer. Who knows?"
Asked if a rotational Taoiseach is a must be on the table as part of any agreement for Fine Gael to go back into government with Fianna Fail, he replied: "Well, look, well over 400,000 people gave their first preference vote to my party in the general election less than a week ago.
"We saw more seats in Dáil Éireann, we saw 23 new TDs elected and we have also received a mandate, a mandate to advance our policies and to play a pivotal and leadership role in government.
"The details are for another day, and indeed are for negotiation and discussion."
Labour will not merely prop up larger parties - Bacik
Earlier, the leader of the Labour Party said that her party will not enter government "merely to prop up" any of the three larger parties following the General Election.
Ivana Bacik made the comments in a statement after the Labour Party held its first parliamentary party meeting since the election.
The party won 11 seats in the 34th Dáil, including seven new TDs. Ms Bacik has said that she will first speak to smaller left-wing parties on government formation, with a view to agreeing a common negotiating platform.
Ms Bacik was due to meet Green Party leader Roderic O'Gorman today, and Labour will hold talks with a delegation from the Social Democrats tomorrow.
In the statement, Ms Bacik said that her priority is to "work to form a progressive, left-of-centre bloc with like-minded parties".
The Social Democrats, which also has 11 TDs following the election, were set to have meetings with Sinn Féin today to explore their options, with Fine Gael "shortly" and the party has also asked to meet with Fianna Fáil.
Ms Bacik earlier confirmed that she has spoken briefly to the leaders of the three larger parties - Fianna Fáil, Fine Gael and Sinn Féin - to tell them that she would initially speak to smaller parties and will have proper discussions with them next week.
Asked whether Labour could go in to a coalition without the Social Democrats, Ms Bacik did not explicitly rule it out, but she insisted the party was solely focused on agreeing a common negotiating platform.
Meanwhile, Social Democrats deputy leader Cian O'Callaghan has said the party was open to cooperating with other parties where there is common ground.
He said it was too early to make a call on whether it would come together with Labour to form a left-wing negotiating bloc.
Mr O'Callaghan added that his party's participation in government was not dependent on any other party going in.
SF's Ó Broin denies voters aren't being offered alternative government
Sinn Féin's Eoin Ó Broin has denied accusations that voters had not been offered a coherent alternative to the Government that has been in place.
Speaking on RTÉ’s Drivetime programme, Mr Ó Broin said that there had been a "huge amount of cooperation" amongst the "progressive parties" over the last number of years.
"If you take, for example, the housing and homelessness crisis and the need for increased social and affordable housing, Sinn Féin, the Social Democrats, Labour, People Before Profit were all part of the very powerful 'Raise the Roof’ coalition led by the trade union movement, organizing protests and demonstrations," he said.
He said that what is "absolutely the case" for the voters who he represents, is that "people cannot afford another five years of Fianna Fáil or Fine Gael."
"Notwithstanding the fact that Fianna Fáil and Fine Gael combined have the largest number of seats, they don’t have a workable majority yet.
"No functioning coalition emerged, in fact, the three parties of Government don’t have enough votes to form another Government."
Mr Ó Broin said that he will be using "every ounce of energy" between now and when a Government is formed to prevent a Fianna Fáil and Fine Gael Government from being formed.
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As Fianna Fáil and Fine Gael move towards opening talks on forming a government, their most likely partners right now appear to be independent TDs.
The newly formed Regional Group has appointed Michael Lowry as the point of contact for any discussions with the leaders of Fianna Fáil and Fine Gael.
The group includes Mr Lowry, Seán Canney, Kevin 'Boxer' Moran, Noel Grealish, Marian Harkin, Barry Heneghan, Gillian Toole and Verona Murphy.
Mr Martin has already said he was heartened by the initial conversations he has had with some Independent TDs.
The Dáil meets on 18 December and it is not expected that a government will be in place at that point.
Mr Harris has appointed a team for talks with other parties and deputies on forming the next administration.
The team is headed by deputy leader Helen McEntee and includes deputies Paschal Donohoe, Jennifer Carroll MacNeill, Martin Heydon, John Cummins and John Paul O'Shea.
In a statement, Fine Gael said that Mr Harris has held preliminary conversations that "were informative and serious" with Fianna Fáil, Labour, the Social Democrats and a number of independent TDs.
Ó Fearghaíl to seek third term
A Ceann Comhairle will have to be voted into office when the Dáil meets.
Outgoing Ceann Comhairle Seán Ó Fearghaíl is to seek re-election to the role.
Mr Ó Fearghaíl had intended to step back from the position, but it is understood that he has been approached by TDs from across the political spectrum to consider retaking the role.
It is likely that Mr Ó Fearghaíl will focus on providing stability and continuity.
He was first elected to the role in 2016, and was re-elected following the 2020 General Election.
As outgoing Ceann Comhairle, he was automatically re-elected to the Dáil for the Kildare South constituency in both the 2020 and 2024 elections.
Analysis: Independents move centre stage in Government formation talks
Additional reporting Sandra Hurley
