Fine Gael and Fianna Fail have taken part in "constructive and cordial" talks at Leinster House about the formation of a new government.
The meeting, held in the Sycamore Room, was attended by the negotiating teams of both parties and lasted an hour and a half.
In statements, both parties described the meeting as "constructive and cordial."
They have agreed to meet again tomorrow.
The meeting follows Fianna Fáil's rejection of a partnership deal last week.
Earlier, Taoiseach Enda Kenny said the general election results required a new way of doing politics to provide a stable and lasting government.
He said the new political reality was difficult for all parties, including his own. He added that the complex result required a big, bold response.
Mr Kenny said many efforts had been made to form a government since the election. He said he hoped negotiating teams would deliver a government in the national interest and for the people
Negotiating teams from both parties held separate meetings this morning ahead of the inter-party talks.
The initial focus of the talks was expected to be how a minority government would work, rather than discussions on policies. A policy document is not expected to emerge from the meeting.
The meeting between the parties came after exploratory talks between the two parties over the weekend.
While Fine Gael insists that Mr Kenny offers the best prospect of leading a stable minority government, the party has not ruled out the prospect of a minority Fianna Fáil government.
The Independents want both parties to reach a written agreement covering a number of budgets before the Dáil reconvenes on Thursday.
TDs are then due to vote to elect a taoiseach for a third time since the General Election just over six weeks ago.
Background: Political Correspondent Martina Fitzgerald
It is understood both teams will comprise four deputies, but this will be confirmed after their respective party meetings.
It will be the second time the two parties have engaged since the weekend.
Fianna Fáil made an overture to Fine Gael and on Saturday a meeting took place between Leo Varadkar and Jim O'Callaghan along with two party advisors.
Both parties have also agreed to calm the situation and tone down public comments.
Sinn Féin to be 'constructive opposition' - Adams
Sinn Féin leader Gerry Adams has said his party will hold Fianna Fáil to account as part of a constructive opposition if a minority government is formed.
Speaking in Belfast, Mr Adams said "It appears this time around Sinn Féin won't be in government and then we have to be a constructive opposition and we will hold Fianna Fáil to account, because no matter how much they want to long-arm it, they will be as much a part of that government as Fine Gael."
Asked if he could see Sinn Féin and Fianna Fáil finding common ground in a future coalition, he said it would be very difficult, and it is impossible to make a call on that because Fianna Fáil have pointedly said they would not talk to Sinn Féin.
Independents wary of short-lived minority government
Two Independent TDs heavily involved in negotiations with both Fianna Fáíl and Fine Gael have sought assurances a minority-led government would last a number of years.
Roscommon-Galway TD Michael Fitzmaurice has said he expects Fine Gael and Fianna Fáil to agree on a way to pass at least three budgets.
He said, however, it was up to the two parties to clarify some issues before Independents would decide who to support.
Kerry TD Michael Healy-Rae said he hoped a sound foundation for government could be reached this week and that once formed it would last a number of years.
Speaking on RTÉ's Morning Ireland, Mr Healy-Rae said as far as he was concerned it was "all open".
"Everybody will readily agree that it won't last for five years but at the same time people don't want a hash of a thing that will be there for six or eight months or something and then collapse at the first hurdle."
