The President can refuse a dissolution request when the Taoiseach does not have the support of the majority of the Dáil
The odds on a second election have shortened after Fianna Fáil has rejected a Fine Gael proposal to enter a partnership.
Should that happen, acting Taoiseach Enda Kenny would have to travel to Áras an Uachtaráin seek a dissolution of the 32nd Dáil from the President.
Michael D Higgins would have powers to refuse the dissolution request - but would that influence the outcome of the political stalemate?
Conor O’Mahony is a constitutional law lecturer in UCC, and spoke to Newstalk Breakfast about the situation.
"It's really quite straightforward, and pretty limited," Conor explained. "The Constitution grants [the President] the absolute discretion - where the Taoiseach does not have the support of the majority of the Dáil - to refuse a dissolution.
"Sometimes you will hear media commentary which suggests it goes further than that, and people describe the idea of the president reaching out... but that's not what the Constitution envisages. The President has the power to refuse a dissolution, and the purpose of that would be to really try and suggest to the various parties that they need to spend a bit longer at it.
"There's no power beyond that for the President to chair those negotiations, or try to bang heads together," he added. "It's really very much a question of the President trying to create the conditions whereby attention might focus a little bit more".
The President's constitutional power has never been exercised, so there is no clear precedent. However, previous presidents have come close.
In 1994, Labour walked out of government, and it is believed that Mary Robinson implied to Albert Reynolds that she would not dissolve the Dáil - a move that may have helped lead to the formation of the Rainbow Coalition.
Conor also explained how in 1987 "there were some back channel discussions between Patrick Hillery and Garret FitzGerald, in which Paddy Hillery suggested he would exercise that power".
However, in both cases the power was not ultimately exercised - if Michael D Higgins does choose to do so, it will be a first.
"In the event that the Taoiseach has a majority, then the President has no choice but to grant a dissolution - that's what we see before every election is called," Conor explained. "It's where the Taoiseach no longer has a majority, as is currently the case, that this discretion arises for the President.
"Really it's a question of him making an assessment as to whether he thinks this might bring about a better chance of government formation. If there's no realistic prospect of that, there's no real point - all it does is prolong the agony," he added.
"What I'm saying is that he may refuse a dissolution - I think it is quite plausible he would refuse a dissolution. Other suggestions - around addressing the Oireachtas or chairing negotiations - I think those things are not going to happen," Conor observed.