Fine Gael and Fianna Fáil say they have agreed the need to form a strong, stable government.
Efforts to form a government are set to continue after it was revealed no new laws can be passed from next week.
The lack of a new government will paralyse the legislative process from midnight on Sunday.
Following talks on Wednesday, Fine Gael and Fianna Fáil released identical statements.
They said: "Both Fine Gael and Fianna Fáil met this afternoon and had 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 a working majority in the Dáil.
"They will meet again over the coming days and will both continue to reach out and engage with other parties."
There are concerns we are heading into a Constitutional crisis after it was confirmed that no new laws can be passed after midnight this Sunday.
The Ceann Comhairle has received legal advice warning that the Seanad will not be able to sit unless a new government is in place.
Seán Ó Fearghaíl confirmed the situation in a letter to Fianna Fáil TD Marc McSharry yesterday.
No new laws can be passed from midnight on Sunday.
The Ceann Comhairle confirmed legal advice in a letter to @MarcMacsharryTD. Proper constitutional crisis
— Seán Defoe (@SeanDefoe) March 25, 2020
He noted that 11 members of the Seanad must be nominated by the Taoiseach and the Upper House cannot meet until they are in place.
Leo Varadkar does not have the power to nominate them in his capacity as acting Taoiseach.
If the Seanad cannot sit, no new laws can be passed – potentially leaving the Oireachtas paralysed amid the COVID-19 outbreak.
“This obviously gives rise to a very serious Constitutional problem,” wrote Deputy Ó Fearghaíl.
“From midnight next Sunday 29th March 2020, the Houses of the Oireachtas will not be able to pass legislation.
“Given the profound implications of this, I am sure that the Taoiseach, party and group leaders will want to address this serious legislative incapacity at the earliest possible opportunity.”
11 members of the Seanad are appointed by the Taoiseach.
Leo Varadkar can't do that as outgoing Taoiseach.
Seanad can't meet without them.
Therefore no new laws as of Sunday.
New Taoiseach & government needed to break deadlock
— Seán Defoe (@SeanDefoe) March 25, 2020
There are sixty members of the Seanad – 49 elected and 11 nominated.
Speaking at a press conference earlier on Wednesday, Mr Varadkar said the situation is cause for real concern but insisted the laws that are needed are being passed this week.
He said he believes it is possible to form a new government in the coming weeks – but noted that a working majority of 82 to 85 TDs will be needed, adding “we’re not there yet.”
With reporting from Sean Defoe and Jack Quann
