Fine Gael and Fianna Fáil announced on Saturday evening that they will meet early next week for talks.
It follows failed attempts by Fine Gael's Enda Kenny and Fianna Fáil counterpart Micheál Martin to be elected taoiseach on Wednesday.
It was their second attempt since February's General Election.
On Saturday evening, acting Taoiseach Enda Kenny left government buildings and a brief statement from both parties followed.
The statements said that negotiating teams from the parties would begin talks early next week.
The focus will be on how a viable minority government would work. Enda Kenny's preferred option has been for an equal partnership government.
Independent TDs were not involved in the talks. Each party will continue separate parallel talks with the Independents.
On Wednesday, Enda Kenny was defeated by 80 votes to 51, while Micheál Martin was defeated by 95 votes to 43.
Mr Kenny insisted at a party meeting on Tuesday he would not consider a short-term deal.
The Dáil last met on 10 March, when TDs failed to elect a taoiseach to form a government.
Following the February election to the Dáil, Fine Gael has 50 seats, Fianna Fáil 44, Sinn Féin 23 and the Labour Party got seven.
The Republic of Ireland's two main political parties have agreed to meet for discussions on how a minority government would work.