Fine Gael and Fianna Fáil have agreed to hold a one-day policy exchange next week, following contact between Leo Varadkar and Micheál Martin earlier today.
The two leaders also agreed to meet again following these discussions.
It is expected that the policies which are likely to be reviewed include housing, climate change, health, cost-of-living, the economy, and taxation.
However, at a meeting of the Fine Gael parliamentary party this afternoon, Mr Varadkar said the party position on government formation has not changed, and he intends to lead them into opposition.
Fine Gael's parliamentary party chairman, Martin Heydon, said in a statement that Mr Varadkar "does not believe there is currently sufficient basis for appointing a negotiating team, or initiating discussions on a potential Programme for Government".
Fine Gael will also meet the Green Party for similar discussions next week.
In a statement, a Fianna Fáil spokesman said this evening: "Fianna Fáil had a very constructive engagement with the Green Party today and will continue that process tomorrow.
"We also had a constructive discussion with the Rural Independent Group. We will participate in the policy exchange next week with Fine Gael and our view is that serious work now needs to be done to create a Programme for Government that will deliver demonstrable change in the areas of health, housing and climate change."
Yesterday, Sinn Féin leader Mary Lou McDonald told a rally in Dublin that Fianna Fáil and Fine Gael are engaged in a blatant and deliberate attempt to keep her party out of power.
She said a coalition of the two would not change things and would instead frustrate the change that people want.
Ms McDonald called on the two parties to engage with Sinn Féin and she said her party respected the mandate of others.
Around 500 party supporters attended the meeting in Liberty Hall.
Meanwhile, the Dáil will not vote on the position of Taoiseach when it meets next Thursday.
The Business Committee of the House, which is made up of representatives of all parties and independents, has agreed that the vote will not take place.
There was no specific proposal from any party to hold such a vote. It was agreed that because there has not been enough progress made in government formation talks, it would be futile to hold the vote.
There will be statements on the European Council meeting and the coronavirus when the Dáil reconvenes on 5 March.
There is a Constitutional requirement for the Dáil to vote on the Taoiseach in its first meeting after an election, but there is no such requirement in relation to subsequent sittings.
Following the last general election in 2016, there was a vote for Taoiseach on the first Dáil sitting and the third, but not the second sitting.
Additional reporting: Mary Regan