All eyes are now on a coalition of Fianna Fail, Fine Gael and the Green Party to form the next Government.
Ireland's largest party Fianna Fail agreed not to go into coalition with Sinn Fein following a four-hour meeting of its parliamentary party on Thursday.
Two of the three largest parties will need to cooperate to form a government, with Fianna Fail on 38 seats, Sinn Fein on 37 and center-right Fine Gael on 35 in the 160-seat parliament.
Fianna Fail Leader Micháel Martin has stated he will not enter Government with Sinn Fein - as has Fine Gael leader Leo Varadkar.
She said it was "disgraceful" the two parties "believe they can set aside the democratic mandate of Sinn Fein".More news: France Reports First Coronavirus Death After 80-Year-Old Tourist Dies
Some Fianna Fail lawmakers suggested the party could lead a minority government similar to the previous administration Mr Varadkar led via a co-operation deal with Fianna Fail, then the main opposition party.
Both Fianna Fail and Fine Gael have reiterated their intention not to go into a coalition with Sinn Fein - which makes forming a Government more hard.
Ms McDonald said forming a coalition without them or Fine Gael would be "very, very tricky".
But on Thursday he told Ireland's national broadcaster RTE News his group would not hold talks with Sinn Fein.
He said: "In terms of the future, I can't be certain how all of this is going to work".More news: USS Normandy seizes 150 missiles bound for Yemen
The minister there was a "stark choice" between a coalition between the two parties and holding another general election.
A source from the centre-right party told the Irish Independent there was "pretty unanimous support" for Mr Martin's decision.
Senior ministers have said Ireland's premier would insist on a number of pledges, including continuing the roll out of broadband in rural parts of the country and improving childcare subsidies.
In Saturday's election, Sinn Fein won 25% of the first preference vote, with the party capitalizing on discontent with a housing shortage and an ailing health service. "The first step of that process is for our leader Mary Lou McDonald to meet with the leader of Fianna Fail, Micheal Martin".
In a statement, she said: "I told Micheal Martin that people who voted for Sinn Fein and those who voted more widely for change are angry that Fianna Fail is denying them the respect of sitting down with the party that represents them".More news: 'Tuchel is PSG's worst coach since the Qatari takeover'