The Social Democrats say they will definitely not support a Fianna Fáil - Fine Gael government.
Micheál Martin and Leo Varadkar are due to sit down for "exploratory talks" on government formation early next week.
With the two men repeatedly having ruled out working with Sinn Féin, they would need to work together to form a government - but would still require additional support from smaller parties or independents.
However, Social Democrats co-leader Catherine Murphy confirmed this morning her party - which has six TDs in the 33rd Dáil - will not back such a coalition.
It is a definite no to FF FG https://t.co/efdmzYcHn1
— Catherine Murphy (@CathMurphyTD) February 22, 2020
Speaking to Newstalk, Deputy Murphy said: "Those two parties really governed for the last four years, and there's a real desire to tackle some of the major issues such as housing.
"We're just working on the issues, and we're not ruling anything out after that. But the one thing we are ruling out is more of the same - and that means the Fianna Fáil or Fine Gael combination."
Members of the party did not support any of the candidates in the votes for the Taoiseach in the Dáil on Thursday, with co-leader Róisín Shortall saying it would be pointless for them to do so until a programme for government was in place.
The party's newly-elected Dublin Central TD Gary Gannon yesterday evening said they wouldn't be in "any way" involved in a 'grand coalition' between Fine Gael and Fianna Fáil.
Writing on Twitter, he said: "I respect the mandate of all parties, but that coalition won’t be one that we will have any part in.
"In addition to this, I’m never going to vote for a person to be Taoiseach without first sitting down to discuss and scrutinise an actual programme for government."
He claimed his inbox was full of people "demanding I vote Sinn Féin and others threatening to never vote for or support me again if I do vote Sinn Féin" - but said his decision on who to vote for "will be about the policy and the vision for an actual change in direction".
I stood on a mandate of change since my first election in 2014. I will ensure through my own contributions and that of my colleagues in the SocDems that this is delivered upon.. but we’ll do it our way and won’t sacrifice our own principles as it is achieved. 💜
— Gary Gannon TD (@GaryGannonTD) February 21, 2020
Sinn Féin leader Mary Lou McDonald yesterday said her party would again meet with the Green Party, the Social Democrats, the Labour Party, Solidarity-People Before Profit, Independents4Change and other independents in the coming days.
Ms McDonald has said she would like to form a left-wing government, although the Dáil numbers have made government formation efforts challenging for all parties.
In a statement yesterday, the Sinn Féin leader accused Fine Gael and Fianna Fáil of seeking a "carve-up of political power that is about keeping them in office and blocking change".