Sinn Fein vice-president Michelle O'Neill has said that the next Irish Government must begin to actively plan for Irish unity.
riting in Wednesday's Belfast Telegraph, she said that could involve appointing a minister responsible for preparing for unity, a citizens' forum on constitutional change, and a green paper on unity.
But DUP leader Arlene Foster has ruled out a border poll and claimed that support for Irish unity is actually declining.
Writing in the Belfast Telegraph, she said: "At the general election eight weeks ago, just over 38% of votes cast were for pro-united Ireland parties.
"That compares with 42% in the 2001 general election, 19 years ago."
Following its stunning election result, Sinn Fein is working to establish the viability of a left-wing coalition government in the Republic.
Party president Mary Lou McDonald will meet the smaller parties and a range of independent TDS on Wednesday to see if she has the numbers to assemble a workable administration.
It remains highly doubtful that Ms McDonald can garner enough partners to reach the magic number of 80 TDs to command a majority in the Dail.
But if she did pull it off, it would represent a landmark moment for the state, as it would be the first time in more than 90 years when either Fianna Fail or Fine Gael was not in power.
Ms McDonald is talking to smaller parties including the Greens, Labour, the Social Democrats, Solidarity/People Before Profit, and a sizeable number of independent TDs.
They would be junior coalition partners in a Sinn Fein-led government. A coalition of Fianna Fail, Sinn Fein and the Green Party - which ended up with 12 seats - is seen as a potentially more realistic option.
Fianna Fail emerged from the election as the largest party by the narrowest margin over the surging Sinn Fein. Micheal Martin's party finished with 38 seats to Sinn Fein's 37 at the end of two days of counting.
But given the Fianna Fail speaker was re-elected without contest, both parties essentially "won" the same number of seats.
Taoiseach Leo Varadkar's Fine Gael was the big loser, recording the second-worst result in its history, winning only 35 seats having entered the campaign as the largest party on 47.
Sinn Fein's Pearse Doherty has been appointed head of the party's negotiating team. He said Sinn Fein was reaching out to the Social Democrats, the Green Party, People Before Profit and Labour.
"We have said consistently we will meet with all parties to look at delivering real change for people," the Donegal TD added. Ms McDonald has predicted she could be the next Taoiseach.
Newly elected Sinn Fein TD for Cavan and Monaghan Matt Carthy acknowledged it would be difficult to reach 80 seats.
"The numbers of that are going to be tight," he told RTE. "But we have an obligation, and Mary Lou McDonald has already set this in train, to talk to the progressive parties that were clearly part of the mandate for change."
There are differing views within Fianna Fail, with some members remaining opposed to any alliance with Sinn Fein and others more open to the prospect.
Fianna Fail's finance spokesman Michael McGrath said the onus is now "first and foremost" on Ms McDonald to demonstrate how much support there is in the Dail for a Sinn Fein-led government without Fianna Fail and Fine Gael.
Fianna Fail recognised that the election demonstrated a demand for major change in Irish politics and that Sinn Fein had performed exceptionally well, he said. Mr McGrath told RTE that Fianna Fail's new parliamentary party would meet tomorrow to assess where it wants to go from here.
He described his party's economic policies and those of Sinn Fein as "poles apart".
Mr Varadkar has maintained his pre-election stance and ruled out any Fine Gael/Sinn Fein coalition.
Another permutation could see Fine Gael and Fianna Fail entering power together in a grand coalition with another smaller party, although that prospect looks increasingly unlikely. Irish President Michael D Higgins said it was "important to be positive" as the political parties enter discussions.
"I have great confidence in the process that we have. We are very fortunate to have a constitution that lays things down very clearly," he said.
"I want to wish all of the people who will be having discussions with each other every success. Whatever they decide, we are obviously facing into a period in which the EU is redefining itself on several levels and we are moving into some of the most complex parts of the trade agreement with our neighbours through Brexit.
"There are huge international issues as well. The themes standing in every background of every politician alive are issues of climate change and loss of biodiversity.
"I wish them every success with the different options that they have. Obviously at different times, at the appropriate time, people will visit me at the Aras (an Uachtarain). I will give my blessing for however long it lasts."
Sinn Fein received 24.5% of the vote share on first preference in Saturday's election, Fianna Fail got 22.2% and Fine Gael 20.9%.
The final breakdown of the 160 seats was Fianna Fail 38, Sinn Fein 37, Fine Gael 35, Greens 12, Labour six, Social Democrats six, Solidarity/People Before Profit five, Aontu one, Independents4Change one and independents 19.