The leaders of Fine Gael and Fianna Fáil will meet next week in exploratory talks on forming a government.
It follows a phone call between Leo Varadkar and Micheál Martin on Thursday evening, RTÉ reports.
On Thursday, Mr Varadkar tendered his resignation as taoiseach (Irish prime minister), after efforts in the Dáil to elect his successor ended without agreement.
There were four nominations for the post.
These included Fine Gael leader Mr Varadkar and Fianna Fáil's Mr Martin, as well as Sinn Féin leader Mary Lou McDonald, nominated by her own party, and Green Party leader Eamon Ryan.
Mr Varadkar will continue as caretaker taoiseach.
He and the previous government will remain in place until a new government can be formed.
Fine Gael agreed on Monday while it could engage in exploratory talks, it would not enter negotiations on forming a new government.
The party has also accepted an offer to have similar talks with the Green Party next week.
Four attempts were made to elect a taoiseach on Thursday, with Sinn Féin leader Ms McDonald receiving the most votes.
However, she was still 45 short of the amount required to form a government.
Sinn Féin emerged from the poll on 8 February with the highest number of first preference votes and it won 37 seats, one seat fewer than Fianna Fáil, which had 38.