Mary Lou McDonald will heap pressure on Micheál Martin to go into government by presenting him with a ready-made coalition.
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Sinn Féin will seek to get the Green Party and Social Democrats on board before talking to Fianna Fáil about forming a government.
Eamon Ryan will be offered the climate change portfolio and Róisín Shortall health, in a multi-party coalition. Sinn Féin will insist it takes the housing portfolio.
The position of Taoiseach would be on the table in talks with Fianna Fáil, the Irish Independent understands.
The move comes as Taoiseach Leo Varadkar cleared the path for Sinn Féin to form a coalition with left-wing parties or Fianna Fáil. “Sinn Féin won the most votes so the onus is on it to try for a government with left-wing parties or Fianna Fáil,” Mr Varadkar told the Irish Independent.
Fianna Fáil is also leaving it to Sinn Féin to seek to put an alternative left-wing coalition together. However, without Fianna Fáil, the other parties don’t have the numbers to achieve a majority.
A string of Fianna Fáil TDs also stated their opposition to government with Sinn Féin.
Labour leader Brendan Howlin will also be contacted – but he has indicated a reluctance to deal with Sinn Féin.
The opening rounds of negotiations will focus on seeking common ground on housing, health and economic policies.
Ms McDonald might yet have the backing of the Greens and Social Democrats to become Taoiseach.
But this would be yielded in negotiations with Fianna Fáil - at a price. The finance minister's job would also be up for grabs between Sinn Féin and Fianna Fáil.
Sinn Féin will appoint a negotiating team for coalition talks as soon as possible.
Senior Fianna Fáil figures said they would allow Sinn Féin to form a government with the other left-wing parties.
"If they want Venezuela or whatever let them go off and put it together," a senior source said, before adding: "We won't be helping them."
Meanwhile, Sinn Féin's ard chomhairle met last night to sign off on its negotiation priorities.
Speaking ahead of the talks, Ms McDonald said she would approach Fianna Fáil and Fine Gael in the coming days - but insisted her preference was for a government without either party.
"In talking to everybody the real litmus test is therefore how much change you can agree and how much change can be delivered."
She said "the mathematics of this Dáil are tricky" and that the "two-party system is broken forever".
The ard chomhairle meeting was attended by a number of senior figures in Sinn Féin who are based in Northern Ireland, including former leader Gerry Adams, deputy leader Michelle O'Neill, party chairman Declan Kearney and the North's Finance Minister Conor Murphy.
Mr Murphy was forced to apologise last week for linking murder victim Paul Quinn to criminality in 2007.
Meanwhile, Fianna Fáil TD Barry Cowen frankly stated that his party did not have a mandate to go into government and suggested that it would be better off in opposition.
"I don't believe 22pc of the vote is a vote of confidence in our ability to govern," Mr Cowen said.
"I don't believe losing a few seats is an indication of success. [Sinn Féin] needs to take its ambition seriously now and ensure we get a government from the left."
Mr Cowen said Ms McDonald and the left should "form a government, find the €22bn and off they go".
A number of senior Fianna Fáil TDs were yesterday agreeing with Mr Cowen's suggestion the party should go into opposition and conceded that this could result in Micheál Martin stepping aside.
Mr Cowen's comments came as it looked like Fianna Fáil was set to see up to 13 outgoing TDs lose their seats.
There were a number of high-profile casualties for the party including Lisa Chambers, Timmy Dooley and Pat 'The Cope' Gallagher.
As the results were coming in one councillor in Cavan, John Paul Feely, said he was "baffled" that Mr Martin was remaining party leader.
"I think if he had any decency he would have stepped aside at this stage," he said.
Meanwhile, smaller parties such as the Green Party and the Social Democrats looked like they would make significant gains when the final votes are counted.
The Labour Party did not look like it would be making major gains.
Solidarity-People Before Profit also looked likely to lose a seat but had a surprise win through Gino Kenny retaining his seat.