Fine Gael and Fianna Fáil have both said they will meet again following talks at Government Buildings.
Today's talks were described by one source as "good" and both parties exchanged papers on the parameters for supporting a minority government.
Further talks are planned for tomorrow.
Fine Gael and Fianna Fáil also held a 90-minute meeting last night on the shape of a new government, with both sides describing the talks as "constructive and cordial".
The respective negotiating teams were headed by Fine Gael's Frances Fitzgerald and Fianna Fáil's Michael McGrath.
Earlier, Fianna Fáil's Éamon Ó Cuív said there was now a clear path and process in place for government formation.
However, Independent TD Mattie McGrath said he believed the country was edging towards a general election unless someone blinked first.
Yesterday, Taoiseach Enda Kenny told a function that the election outcome required a different kind of answer and a big bold response from the parties.
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.
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 nearly seven weeks ago.
Analysis: Political Correspondent David Davin-Power
As Fine Gael and Fianna Fáil were holding talks in Government Buildings, Taoiseach Enda Kenny was attending an event at the National Concert Hall nearby.
Mr Kenny told those gathered at the function that the election outcome required a different kind of answer and a big bold response from the parties.
When that 'response' will be finalised is unclear but it is unlikely to be this week.
Once complete it will include a new kind of relationship between government and opposition, and a new kind of Dáil where business will be ordered by agreement and not by government diktat.
By Thursday and the next vote for taoiseach, Fine Gael will be hoping that some Independents will be persuaded to support Mr Kenny and give his push for leadership some badly needed momentum.
