Fianna Fáil and Fine Gael say they held "a good constructive first meeting" on the Confidence and Supply Arrangement on Thursday.
Both parties have agreed to meet again next week.
They will exchange papers next Tuesday, October 30th and these will be reviewed.
This will be followed by a full plenary meeting next Thursday, November 1st.
Negotiating teams are reviewing the deal that has kept the Government in power over the past three budgets.
The teams will then discuss a potential extension.
Tánaiste Simon Coveney, Finance Minister Paschal Donohoe, Social Protection Minister Regina Doherty and Parliamentary Party chairman Martin Heydon are representing Fine Gael.
Meanwhile, the Fianna Fáil team is led by deputy leader Dara Calleary, finance spokesperson Michael McGrath, Brexit spokesperson Lisa Chambers and agriculture spokesperson Charlie McConalouge.
An identical statement from both parties says they agreed not to do media doorsteps after the meetings.
So Fine Gael's negotiating team for the Confidence and Supply deal called a press event for 5pm... It's now just been cancelled... Read into that what you will
— Sean Defoe (@SeanDefoe) October 25, 2018
Both sides in the talks have already pledged not to bring down the Government while Brexit remains in the balance – and indicated they are entering negotiations without pre-conditions.
Fine Gael is seeking a two year extension to the agreement with an agreed election date in 2020.
However, Fianna Fáil has been reluctant to accept that – with the outcome of the review to inform the progress of negotiations.
The Taoiseach has said he sees no reason the talks cannot be completed in a matter of weeks, however Fianna Fáil has been more cautious in its approach.
The prospect of an election before Christmas still looks remote – but the next few weeks will decide how long this government will last.
Speaking earlier, Leo Varadkar said he is pitching for a two-year extension of the deal to keep his party in power.
He said: "The priority from our point of view - and our objective - is to make sure the country has a stable Government and a functioning Government, that could provide good Government to citizens over the next number of years.
"As I said in the letter that I wrote to Micheál 53 days ago now, a Government can't really function if it doesn't know if it's going to be around next week or next month."
Additional reporting by Sean Defoe, Stephen McNeice and Jack Quann