The Green Party has called on all parties to suspend government formation talks and to "work towards forming a crisis national government" to deal with the coronavirus outbreak.
The Greens proposed that such an arrangement be reviewed in three months.
The call came after a meeting of the parliamentary party today.
The Social Democrats are not going to support the proposal.
Replying to a letter from the Greens, the Social Democrats say they are already engaged in a cross-party process with senior public health officials and other senior departmental figures, which is due to meet again on Friday.
The Social Democrats say they believe this "remains the best forum within which to formulate an effective national response to the enormithy of the crisis facing the country".
Informal talks on government formation between the deputy leaders of Fine Gael and Fianna Fáil have concluded.
Simon Coveney and Dara Calleary were engaging in a scoping exercise on how negotiations on a programme for government can be conducted.
Fine Gael sources said it was a "good constructive meeting", adding that initial talks between members of the negotiating teams will follow.
Yesterday afternoon, Fine Gael leader Leo Varadkar and Fianna Fáil leader Micheál Martin agreed that their two parties should commence detailed coalition talks.
Fianna Fáil negotiator Michael McGrath said today's engagement with Fine Gael was "really constructive".
He said that the "full negotiating teams" would meet tomorrow and then "get into more detail".
Mr McGrath said he anticipated tomorrow's talks would focus principally on how to tackle the coronavirus.
The Cork South Central TD said it was "difficult" to put a timeframe on how long the discussions with Fine Gael would last.
Fine Gael negotiator Paschal Donohoe said he believed in-depth, but preliminary, talks could get under way by Sunday.
Mr Donohoe said: "At the best of time's this would be a very demanding political exercise, but we are truly in extraordinary times and that will influence the discussions we will have with political colleagues all over the spectrum of Irish political life.
"From my experience of these discussions I know it is not helpful if others hear about my priorities from the media."
He added: "We have a duty however to assess our mandate with the parliamentary party ahead of a vote for Taoiseach next Thursday."
Formal talks cannot commence until Fine Gael's parliamentary party gives its approval next week.
Significant progress towards new government - now what?
Independent TD Catherine Connolly said she was not surprised at the developments, but contended that Fianna Fáil and Fine Gael were never going to accept the message from the people that fundamental changes were required.
The Galway West TD also claimed that the two parties did not have what she termed "the excuse" to go into coalition up to now, but now do on account of the coronavirus outbreak.
Additional reporting Maggie Doyle
