Fianna Fail and Fine Gael began talks yesterday that are likely to last for a fortnight over a programme for government.
The two parties issued a joint statement after what was described as a "productive" meeting.
"They both agree the need to form a strong, stable government that will help Ireland recover post Covid-19. They are working to develop a programme for government that provides stability and majority support in the Dail," the statement said.
Meanwhile, Green TDs are divided over whether the party should enter government with Fianna Fail and Fine Gael.
Party sources say leader Eamon Ryan is in favour of the Greens ultimately doing a deal with the Civil War parties if efforts to form a national unity government fail.
Carlow-Kilkenny TD Malcolm Noonan said that if a unity government cannot be formed, "the only realistic scenario for a stable government is Fianna Fail and Fine Gael", adding he was "more than willing to talk to them".
However, several new TDs denied the Greens are open to talks with the parties after a senior source said a coalition with Fine Gael and Fianna Fail cannot be ruled out, if a unity government is not possible.
Finance spokesperson Neasa Hourigan said the Greens want to be in government "but the Covid crisis is not the time for a programme for government talks".
"We're focusing all our energies on putting forward a reasonable and palatable proposal for unity government," she said.
Ms Hourigan's view was backed by TDs Roderic O'Gorman and Patrick Costello, who said: "We're asking people to pull together as a country, why aren't politicians doing the same thing?"
Mr Ryan is said to be among those who is enthusiastic to return to government but he is facing resistance from newer members of the parliamentary party, Green Party sources said.
Mr Ryan did not respond to a request for comment.
Mr Noonan denied there is a split but admitted there was a "difference of opinion and it may be significant".
He said anything negotiated would ultimately have to be voted on by members.
"Fine Gael and Fianna Fail is the only option for a stable government. It's bound to be the case there are differences of opinion about the strategy," he said.
Meanwhile, Ceann Comhairle Sean O Fearghail has warned that politicians will be unable to pass any new laws after this Sunday, creating a "very serious constitutional problem".
In an email to Fianna Fail TD Marc MacSharry, Mr O Fearghail confirmed, based on legal advice, that the new Seanad, to be elected next week, could not meet until 11 people are nominated to the upper house by a Taoiseach elected by the current Dail.
"This obviously gives rise to a very serious constitutional problem: from midnight March 29, the Houses of the Oireachtas will not be able to pass legislation," he said.
However, senator and former attorney general Michael McDowell said whether or not the Seanad could sit in the absence of the 11 Taoiseach's appointees had been considered in 2016.
At that time, the legal advice was that it could operate.
Taoiseach Leo Varadkar is putting pressure on the Greens, Independents and small parties, to enter government talks.
He said a government will need to last for the next four years and have "a stable working majority", requiring at least 10 more TDs, as well as those from Fianna Fail and Fine Gael.