The Taoiseach and the Ceann Comhairle have both expressed concern that the Oireachtas will not be able to pass legislation after next weekend.
This is because a new Seanad cannot be fully formed without the 11 Senators nominated by a taoiseach elected by the new Dáil.
Leo Varadkar described this as a "real concern."
Ceann Comhairle Sean Ó Fearghaíl said the matter gives rise to a very serious constitutional problem.
In a letter to Fianna Fáil TD Marc MacSharry, he stated that the profound implications of this would, he is sure, be addressed by party leaders.
Many believe this legal lacuna could inject urgency into government formation talks as Fianna Fáil and Fine Gael meet today.
Mr Varadkar also said it is possible to form a government in a few weeks but it would require ten or more TDs from outside Fianna Fáil and Fine Gael.
He said any new government would need a working majority that would last.
The Fine Gael Leader also indicated that other parties and TDs would be approached after the conclusion of the Fine Gael and Fianna Fáil talks process.
The Green Party has again stated that it would pursue a national unity government and not a coalition with just Fianna Fáil and Fine Gael.
We need your consent to load this rte-player contentWe use rte-player to manage extra content that can set cookies on your device and collect data about your activity. Please review their details and accept them to load the content.Manage Preferences
The negotiations between Fianna Fáil and Fine Gael will be the first discussions since the Fine Gael parliamentary party gave the green light to the process last Thursday.
The Fine Gael delegation will include Simon Coveney, Paschal Donohoe, Heather Humphreys and Hildegarde Naughton.
The Fianna Fáil team includes Dara Calleary, Michael McGrath, Barry Cowen, Anne Rabbitte, Darragh O'Brien and Thomas Byrne.
Yesterday, Mr Varadkar and Fianna Fáil leader Micheál Martin said, in separate interviews, that political efforts recently were correctly focused on measures to tackle the Covid-19 pandemic.
This morning Mr Martin has said a new government must be formed within "a number of weeks."
He said he did not think the current situation is tenable.
Speaking on Newstalk, he said the country needed a government with a solid majority.
Additional reporting Paul Cunningham