A meeting between Taoiseach Leo Varadkar and Fianna Fáil leader Micheál Martin has been described as "constructive" by their respective parties.
The two men met this evening, amid growing speculation that an early general election is increasingly likely.
A Government spokesman said the talks between Mr Varadkar and Mr Martin lasted for an hour and were cordial and constructive.
Mr Varadkar and Mr Martin discussed "Dáil numbers and possible legislative proposals which could be passed in a further Dáil session", according to statements issued by Fine Gael and Fianna Fáil.
The two men are to meet again next week.
The Taoiseach, Leo Varadkar, and the Fianna Fail leader, Micheál Martin, say they had a "constructive meeting" in Dublin on a possible deal for the next Dáil term.
In a statement, they said they "discussed Dáil numbers" - a reference to the difficulty that the Fine Gael-led minority is facing getting legislation passed due to its reduced majority.
The two leaders say they also talked about "possible legislative proposals which could be passed in a further Dáil session" - in other words, a shortlist of priority bills which could be passed before the general election.
The statement, which was issued before 8pm, does not make any reference to the possible date of a general election should an agreement be reached between the two men.
Should they fail to reach agreement, it is widely expected that an election will be held in early to mid-February.
The statement concludes by saying Mr Varadkar and Mr Martin have "agreed to consider matters further and to meet again next week".
Earlier, Mr Varadkar was unable to give an absolute guarantee that the Dáil will reconvene next Wednesday, amid speculation that there could be a snap election called.
Speaking after a Cabinet meeting in Marino in Dublin, Mr Varadkar said that he would have to speak to Fianna Fáil, Independents and other parties over the next few days.
He said the return of the Dáil was planned for Wednesday, before adding that this is case unless something changes.
Mr Varadkar said he does not believe in the Dáil being wound down and if it is to continue it would need to pass more priority legislation.
However, he said that Government could continue without a pledge from Fianna Fáil to go a step further than the Confidence and Supply Agreement by supporting the minority administration.
Independent Alliance Minister Finian McGrath said he remained optimistic that the Government could survive until Easter.
Minister for Health Simon Harris said that Mr Varadkar, and not an Independent TD, would decide the timing of the next election.
Mr Harris was speaking after Rural Independent TD Michael Collins confirmed that he was seeking to table a no-confidence motion in the minister later this month.
Fianna Fáil deputy leader Dara Calleary said that as of now, he assumed the Dáil would resume next week to conduct its business as planned.
Speaking on RTÉ's News at One programme, he said the focus in the coming days had to be on getting agreement by the Northern Ireland parties to restore power-sharing.
Mr Calleary said Fianna Fáil had worked hard to provide stability through the Confidence and Supply Agreement with Fine Gael.
He also that the Taoiseach had been trying to change the goal posts of that agreement.
The Cabinet meeting focused largely on disability issues.
There was also due to be a revision of the medium-term economic growth forecasts now that a more orderly Brexit looks likely.
It is understood that a portion of any extra revenue generated will be set aside to mitigate against expected falls in future corporation tax receipts.