Taoiseach Leo Varadkar and Fianna Fáil leader Micheál Martin met on Wednesday to discuss the future of the confidence and supply deal, sources said.
It is believed that Mr Martin is willing to extend the deal which underpins the Fine Gael-led minority government - possibly for a year - albeit pending further negotiations and conditions.
A Fine Gael statement last night said the pair had discussed the Government’s future and Brexit. They will meet again in the coming days.
Members of Fine Gael and Fianna Fáil have been meeting in recent weeks to review the deal which has propped up the minority Government for the best part of three years.
The deal will reach the end of its lifespan once votes to secure the passage of next year’s budget through the Oireachtas are complete as Fianna Fáil had agreed to vote with the Government on three budgets.
Mr Martin told the Fianna Fáil parliamentary party meeting on Tuesday that the review of the confidence and supply deal to date had identified issues in the areas of health and housing, but that Brexit was a serious issue which was causing public anxiety.
He told TDs and senators that Fianna Fáil has come a long way as a party, and is seen as responsible and trustworthy.
Those present took what Mr Martin was saying as a strong indication that he intends to extend the confidence and supply deal, although he did not specifically say so. Another indication was a warning from the party leader that the coming period will be challenging for Fianna Fáil.
The Cork South Central TD said he would put a number of issues to Mr Varadkar when they meet, and added that the Taoiseach needs to pay attention to Fianna Fáil’s needs.
One TD said: “He said he is clear in his own head that he knows what is in the best interests of the party and the country.”
Mr Varadkar had previously asked Mr Martin to agree to hold the next general election in the summer of 2020.
Mr Martin then wrote to Mr Varadkar suggesting that both agree not to collapse the Government until at least March, when the European Parliament was expected to formally ratify the Brexit withdrawal agreement.
Mr Martin told his TDs on Tuesday that his letter to Mr Varadkar had anticipated volatility in the Brexit process but added that the volatility had some sooner than expected.
He said consistency and stability is important for Ireland, and that the confidence and supply agreement had provided that. Mr Martin also contrasted the stability in Dublin with the current political crisis in London.