Sinn Féin leader Mary Lou McDonald has spoken to Labour and the Social Democrats in a bid to form a left-leaning Government.
As the makeup of the incoming Dáil becomes clear, Labour's Ivana Bacik has also approached the Green Party and the Social Democrats about a centre-left negotiating bloc.
More "intensive engagement" is expected once parties have taken stock of the election results, a spokesperson said.
However, a coalition of Fianna Fáil, Fine Gael, and a third element — possibly made up of Independents or another smaller party — is still seen as the only viable option.
Abolishing the Carer's Allowance means test, improving access to healthcare, and addressing the housing crisis are among the key demands being put forward by a number of Independents who are now likely to be contacted in the coming days as Micheál Martin and Simon Harris seek to gather together a coalition.
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Among those expected to be called are Seán Canney and Kevin 'Boxer' Moran, who both previously served in Government as part of the Independent Alliance.
Marian Harkin and Verona Murphy have also indicated that they would be open to talks, while it is expected Michael Lowry and Noel Grealish will be willing to continue to lend their support to the Government from the opposition benches.
Labour appears to be divided on whether entering a coalition is now the best option after claiming 11 seats.
After being re-elected in Tipperary North, former Labour leader Alan Kelly said: "There's no point standing on the sidelines, you've got to consider going on the pitch to play."
Party colleague Ged Nash said people need to "reflect" on the message given by the electorate ahead of negotiations.
“If you want to make this country better, there’s one way of doing that and that’s to be in Government,” he said.
However, one Labour source said that the party's new TDs are cool on the idea of Government, while another questioned if such a proposal would pass a vote of members.
"There's very few in the parliamentary party who actively want to go into Government," they said.
There is a sense within the party that the cost of going into Government would be a repeat of the 2016 electoral wipeout which followed the coalition with Fine Gael.
Given that the two main parties will be close to the 88 seats needed for a majority, Independents are seen by some as an easier option as this would not require the same level of concessions on policy or ministerial positions that doing a deal with a smaller party would require.
Stating that the means test on the Carer's Allowance would have to be brought in as part of next year's budget, Mr Canney said: "We'd love to be able to go to the wedding, but we have to wait for the invite first."
Meanwhile, Mr Lowry stated that he does not have any red lines. However, he would be expecting delivery for constituents in Tipperary if he agrees to support a coalition from the opposition benches.
Wexford's Ms Murphy said: "I don't deal in red lines, but the obvious commitments made by the Government around abolishing the carer's means test need to be carried out in full."
Without detailing specific demands, Ms Harkin said that she would be happy to be involved in talks around forming a new Government.
"I ran in the hope that I could influence outcomes and that would be the best way to do it," she said.
Poll topper Michael Healy Rae has also said that he is willing to consider everything when it comes to approaches from Fianna Fáil and Fine Gael, but would be expecting delivery on health services in Kerry.
Newcomer Barry Heneghan, who was strongly backed by former Independent Alliance TD Finian McGrath, said he is open to coalition talks, stressing that he ran for election to be a voice for the younger generation.
The first ever 100% Redress party TD Charles Ward has said, without hesitation, that he would be "very interested" in talking with Fianna Fáil and Fine Gael.
It comes as Sinn Féin made contact with Labour and the Social Democrats, but the party has conceded that both Fianna Fáil and Fine Gael have the numbers for a Government.
“The numbers are there for Fianna Fáil and Fine Gael to form a government together. This would be the worst possible outcome for the people of the country,” Sinn Féin's Eoin Ó Broin said.
A spokesperson for the Social Democrats said the party wants to go into Government and said repeatedly that they “will talk to everyone in order to achieve that”.