The Taoiseach has said his party wants to avoid an election and continue the confidence and supply agreement for two more years.
Leo Varadkar was responding to a letter from Fianna Fáil leader Micheál Martin, who requested that Fine Gael and Fianna Fáil commit up front that they will not bring down the Government.
In his letter Mr Martin said that in light of recent developments, and in the run up to the critical period on Brexit, an election would create dangerous instability.
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In response, Leo Varadkar has said the Government's overriding priority at this point is securing a good Withdrawal Agreement in the Brexit talks and preventing the return of a hard border on the island of Ireland.
Mr Varadkar said the best way to guarantee there will not be an election is to start talks to renew the Confidence and Supply deal, and take them forward promptly.
The Taoiseach wants to extend the current agreement to continue providing stable government until at least the summer of 2020.
He said he looks forward to meeting Mr Martin next week to take these matters further.
Mr Martin wrote: "In light of recent developments and as we head into this critical period on Brexit I think it best if we both state upfront, irrespective of what happens during the supply review process, that we both agree not to bring down the government."
He said both parties should continue to pass the Finance Bill and other legislation, at least until a deal was reached between the UK and the EU on Brexit and it had been ratified by Westminster and Strasbourg.
EU leaders are due to meet to discuss the Brexit negotiations at a crunch EU Council summit in Brussels next week.
Fianna Fáil and Fine Gael entered into a Confidence and Supply agreement in May 2016 which enabled the formation of the current minority, Fine Gael-led government.
The agreement is now up for review.
At the end of August, Mr Varadkar wrote to Mr Martin, seeking a two-year extension of the deal. He was rebuffed, with Mr Martin saying he would only begin negotiations after the Budget.
On Tuesday, with the Budget announced, Mr Martin and Mr Varadkar spoke by phone to begin the process of reviewing that deal.
Speaking on RTÉ News after that conversation, the Taoiseach said if the two parties engaged seriously they could have it done by the Halloween break.
He said they would first engage at advisor level, before appointing negotiating teams.
Mr Varadkar also reiterated that his preference was to have a general election in the summer of 2020.