Update: Mary Lou McDonald has said it would be "farcical" to suggest that a Fine Gael - Fianna Fáil government represents the change people voted for.
The Sinn Féin President has hit out at what she called the "old boys club" for excluding her from government formation talks.
It comes after both parties said they would not go into government with Sinn Féin.
Ms McDonald said they are not respecting what the electorate voted for, saying: "Any suggestion that Fine Gael and Fianna Fáil together represents change is farcical, transparently farcical.
"I think it is actually quite disgraceful that the old boys club of Fianna Fáil and Fine Gael believe that they can set aside the democratic mandate of Sinn Féin."
The two main parties have made it clear that policy differences with Sinn Féin mean it is unlikely that they will be able to form a government with them.
Fianna Fáil's Dara Calleary said his party leader Micheál Martin would meet Sinn Féin leader Mary Lou McDonald but was “very clear” that they wouldn’t go into a coalition together.
Policy issues “are too incompatible” between the parties and Fianna Fáil was “unified” on the stance, he said.
Mr Calleary said Fianna Fáil had other issues with Sinn Féin “in terms of the way the party is run” as well as policy differences.
“There are too many bridges,” he added.
Mr Calleary said Fianna Fáil will make an approach to Fine Gael “at some stage”, adding that the outgoing government party has yet to meet as a parliamentary party. That meeting is to take place on Monday.
In the meantime, Fianna Fáil is establishing its negotiating team and “reaching out to other like-minded parties”, he said.
Meanwhile, Fine Gael Minister for Business, Enterprise and Innovation Heather Humphreys said “there is no point” talking to Sinn Féin because both parties wouldn’t be able to agree a programme for government.
The policies of both parties are “absolutely miles apart”, she said.
“It is up to those who won the election to form a government and, in this case, Sinn Féin won the popular vote and Fianna Fáil won the most seats,” Ms Humphreys said.
The Cavan Monaghan TD declined to rule out going into government with Fianna Fáil or agreeing to a confidence and supply arrangement propping up Fianna Fáil ahead of her party’s parliamentary meeting after the weekend.
Sinn Féin has admitted it cannot form a stable government without either Fianna Fáil or Fine Gael, who it accuses of “arrogance” and “recklessness” for refusing to talk to the party about a coalition.
Eoin O Broin, one of Sinn Féin's government formation negotiators, said Fianna Fáil leader Micheál Martin is doing the “most irresponsible thing” by snubbing Sinn Féin's voters and suggesting a new grand coalition government that was rejected at the ballot box.
Asked how Sinn Féin can form a government without either Fianna Fáil or Fine Gael, he said: “You can’t.”
Mr O Broin said: “The most irresponsible thing to do is what Micheál Martin has said - to say he won’t talk to a party that now represents 24% of the electorate.
Mr O’Broin said political parties refusing to talk to others and who are “threatening” elections “don’t just smack of arrogance but I actually think there is a certain recklessness.”
People “can’t wait” for the housing and health crisis to be resolved, he told RTÉ’s Morning Ireland.
“It is very clear that the only stable government is going to involve two of the larger parties,” he added.
Despite “huge policy differences” between Sinn Féin and Fianna Fáil, the “only responsible thing to do for any party is to sit down with and to talk” with all other parties, Mr O Broin said.
The Dublin Mid-West TD said Sinn Féin was still awaiting a “formal response” from Mr Martin ruling out coalition talks, but suggested the Fianna Fáil leader may yet change his mind.
“If last week Micheál Martin was saying he wasn’t going to talk to Fine Gael and this week he saying he is, then very clearly Micheál Martin changes his mind on these matters,” he added.
By Digital Desk
Talks aimed at putting together a government will continue today.
People Before Profit will meet the Green Party leader Eamon Ryan and Rise TD Paul Murphy, to try to form an alliance.
Yesterday, Fianna Fáil ruled out going into government with Sinn Féin, but said it would talk to Fine Gael.
Micheál Martin has refused to rule out a second general election as the parties scramble to find a working majority.
Following a meeting with Deputy Martin last night, the Social Democrats said: “We had a cordial and respectful conversation but our concern is it didn’t sound a lot like change.”
Sinn Féin TD Martin Kenny is calling on Fianna Fáil to reconsider its stance on working with his party.
Mr Kenny said: "It's displaying a certain immaturity on behalf of Fianna Fáil in that they need to grow up and recognise that there has been an election, that they and fine Gael both lost quite a number of seats.
"Other parties gained seats and the parties that gained seats are parties that were seeking a different way of doing business and a different type of government and a changed government."
He said they have displayed a "certain disregard for the opinions and the views of the general public out there.”
“Fianna Fáil needs to sit up and listen and be prepared to accept that they will have to be part of providing at least some sense of hope for people out there who want a change.”