Just two seats remain to be decided in the Irish election, both in Longford-Westmeath.
Fine Gael remains the largest party with 49 seats - five ahead of its main rival, Fianna Fáil.
Fianna Fáil leader Micheál Martin has said a new government should not be formed before reforms of the Dail (parliament) take place.
"We believe that the new Dáil should not represent more business as usual - that it should involve a decisive move towards a reformed politics," he said.
The current taoiseach, Enda Kenny, admitted over the weekend that the Fine Gael/Labour coalition government he led for the past five years had failed to secure a return to office.
'State of flux'
He will continue to lead the Republic of Ireland in a caretaker capacity until a new coalition is agreed.
It is now likely that Mr Kenny will be one of three party leaders who could be proposed as a potential Taoiseach when the parliament (Dàil) reconvenes on 10 March.
The other candidates are Fianna Fáil leader Micheál Martin and Sinn Féin president Gerry Adams.
Sinn Féin is currently in third place with 23 seats while independents and smaller parties have also done well with 34 seats between them so far.
However, the junior coalition partner, Labour, has suffered badly with only six seats so far, after winning 37 in 2011.
Mr Kenny - who has been re-elected in Mayo - said it was clear the existing government would not regain power.
He refused to discuss possible options for the next government.
However, he added: "As taoiseach I have a duty and responsibility to see how best we might be able to put together a government."
On Saturday, Fianna Fáil leader Micheál Martin said he was "very pleased" but that it would take time before the shape of the new government becomes clear.
Sinn Féin president Gerry Adams said the election represented a "fundamental realignment of Irish politics".
Labour Party deputy leader Alan Kelly, who narrowly retained his seat in Tipperary, said Fianna Fáil and Fine Gael should "cop themselves on now" and form a government.
"All of this pretending that there are massive issues between them is rubbish. They need to come together, work together and put a government in place for the good of the people."
More than three million people were entitled to vote in Friday's poll, which will return 157 members of parliament, known as TDs. The speaker is automatically returned.
The campaign was fought mainly over economic issues, with the government parties asking voters for their support to keep the recovery going at a time when international storm clouds were gathering.
TDs are being elected according to the single transferable vote system, in which candidates have to reach a quota, before their surplus votes are distributed to others.