The leaders of Fine Gael and Fianna Fáil have agreed to meet early next week for exploratory talks.
It follows a phone conversation between Leo Varadkar and Micheál Martin yesterday evening.
Fine Gael has also accepted an offer from the Green Party for talks next week.
Fine Gael agreed last Monday that the party could engage in exploratory talks but not in programme for government negotiations.
Mr Varadkar tendered his resignation as Taoiseach to President Michael D Higgins last night, but will continue in the role in a caretaker capacity.
Mr Varadkar said that no major policy decisions and financial allocations would be made by the Government from this point on, and the Government would continue to serve until a new administration is formed.
He once again insisted that the onus was on others to put a new government together.
However, he said his party would stand by the State and talk to other parties if there was no other alternative.
Speaking in Brussels at a summit on the next seven-year budget for the bloc, Mr Varadkar said Fine Gael would be willing to discuss forming a new government only on the basis of respect for the 450,000 voters who voted for the party.
He said: "We are willing to talk to other parties about participating in government if the opposition fails to put together a government, [but] that's not something we're planning to do or looking for.
"It could really only ever happen on the basis that there is full respect for the fact that we did win 450,000 votes in this election.
"We have 35 seats. We didn't win this election, but we have almost as many seats and votes as Fianna Fáil and not that many fewer than Sinn Féin, but I don't see any evidence yet that that's not recognised by any other party."
Mr Varadkar said he would not be able to predict the future formation of a government.
On next week's meeting with Mr Martin, he said: "At that meeting we'll discuss ways forward and I will then report to the Fine Gael parliamentary party, probably Wednesday or Thursday of next week, as to whether there is any basis for any further discussions."
Mr Martin has said a new government must be formed that would prioritise State intervention to improve key services.
Speaking in the Dáil yesterday, Mr Martin was again highly critical of Sinn Féin.
Mary Lou McDonald responded by saying that Fianna Fáil and Fine Gael had run the show for the last century and were not willing to let go.
The Dáil made four attempts to elect a taoiseach last night with the Sinn Féin leader receiving most votes.
However, the 45 votes secured by Ms McDonald was well short of the required majority to form a government.
The Dáil will meet again in just under a fortnight.
A new government will most likely take a little longer to emerge though.
Additional reporting Tony Connelly