Update 12 noon: Acting Justice Minister Frances Fitzgerald has said Fine Gael and Fianna Fáil could sit down this weekend to discuss the formation of a government, before talks with Independents have concluded.
Fine Gael has been talking with Independents and smaller parties this week, with Independents to begin roundtable talks with Fianna Fáil on Monday.
Minister Fitzgerald said: "We really do have to sit down and look at what a stable government looks like. Talks between Fine Gael and Fianna Fáil could begin this weekend."
Pressure is mounting on Independent TDs to choose who they will support in the formation of a new government.
Fianna Fáil leader Micheal Martin has said he will talk to Enda Kenny again next week after the Dáil makes another attempt to elect a Taoiseach on Wednesday.
Fine Gael claim Enda Kenny wanted to meet Mr Martin again before that vote, as soon as today, but say Fianna Fáil snubbed a text invite to the discussions.
Talks between the country's two largest parties would be seen as long-awaited progress towards forming a new coalition.
Mr Martin rang Mr Kenny yesterday, the first known contact between the two since the election at the end of February.
Speaking to reporters in Leinster House, Mr Martin said that during the 20-minute conversation, Mr Kenny confirmed the vote to elect a Taoiseach would take place as scheduled on Wednesday, and that he (Mr Martin) then said the pair should meet afterwards to decide on how to move forward.
“We both agreed we would engage in the aftermath of that, and following conclusions on Wednesday,” he said.
Fianna Fáil and Fine Gael were last night in disagreement over who first brought up the subject of talks as they attempted to take control of a key stage in the negotiations narrative, with Fine Gael saying its offer to meet today or this weekend still stands.