Fianna Fáil leader Micheál Martin has said that he would form a coalition with Fine Gael, subject to agreement on policies.
However, several of his party's TDs have questioned the Fianna Fáil stance on Sinn Féin at a private meeting.
Fine Gael deputies and senators were told that the party held good policy discussions with both the Green Party and Fianna Fáil in recent weeks and that leader Leo Varakdar would speak to other party leaders soon.
Overall there was no substantial progress made today on Government formation.
There is the clear signal though that Fianna Fáil is putting all efforts into a possible coalition with Fine Gael.
Fine Gael is publicly playing it cool but, privately, many of its TDs admit that such a coalition is beginning to look more likely.
Sinn Féin accused both parties of trying to lay a pathway for a government that would fly in the face of the change the electorate voted for.
Elsewhere, a Fianna Fáil TD has called for a root-and-branch overhaul of the party following a disappointing election result.
Tipperary Deputy Jackie Cahill warned his colleagues that without major change the party could face a drubbing at the next election, whenever it is called.
Other TDs, including Robert Troy, are understood to have questioned the party's opposition to talking to Sinn Féin about government formation.
Mr Martin told the meeting that he had been given a mandate just a fortnight ago to pursue this course of action.
Meanwhile, the Regional Dáil Technical Group has secured a commitment that there will be a vote for Taoiseach when the house meets again on 19 March.
The group, which has nine TDs, said it hoped this would focus the minds of Fianna Fáil, Fine Gael and Sinn Féin about the urgent need to form a stable and effective government.