Fine Gael and Fianna Fáil are facing into a week of further contacts with smaller parties and Independent TDs to assess the possibility of forming a government.
Since the failure of TDs to elect a new Taoiseach when the 32nd Dáil convened for its first session on Thursday, no grouping has yet presented itself as capable of commanding a majority in the House.
The Dáil will resume on Tuesday week and talks on the formation of a new government will be held before then.
However, commitments by acting Taoiseach Enda Kenny, including St Patrick's Day engagements and an EU summit, will curtail the time he has available for talks over the next nine days.
Mr Kenny, is cutting short a visit to Washington to have more time to devote to the task of putting together a new government.
A team of Fianna Fáil negotiators are to meet with Independents and smaller parties over the coming days to talk about government formation.
The team consists of four TDs: finance spokesman Michael McGrath; spokesperson for education Charlie McConalogue; environment spokesperson Barry Cowen; and newly-elected TD for Dublin Bay South Jim O'Callaghan.
Mr McGrath has said the negotiating team plan to meet with the Social Democrats and other Independents on Tuesday.
He said the talks will be framed on forming a minority Fianna Fáil government led by Micheál Martin.
He said the feeling in the party is to work towards forming a government and avoiding the need for another general election.
The separate negotiations being undertaken by Fine Gael and Fianna Fáil are taking place against the backdrop of an opinion poll this weekend which suggests there would only be minor changes to the general election outcome if voters were asked to go to the polls again.
Mr Cowen has said his party is ruling out the prospect of a grand coalition with Fine Gael.
Speaking on RTÉ's The Week in Politics, Mr Cowen said his party is working on the basis of a mandate for change, adding that it is what the electorate voted for.