Left-leaning opposition parties are due to meet today to continue discussions over running a potential combined presidential candidate.
The Social Democrats, Labour and People Before Profit-Solidarity confirmed the meeting during media engagements at Leinster House.
While Social Democrats TD Gary Gannon said his party does not currently have a preferred candidate, he believes both Independent TD Catherine Connolly and Independent senator Frances Black are so far the only two people currently on his shortlist.
He responded to a question over whether he would support Sinn Féin leader Mary Lou McDonald running as a candidate by saying "I'm indifferent to it".
On Monday, Sinn Féin said that it was not ruling in or out a presidential run for Ms McDonald.
Asked if he has a preferred candidate, Labour TD Ged Nash said he believes his party leader Ivana Bacik would be a strong contender, before saying that is unlikely to happen.
He rejected suggestions that he mentioned his party leader's name in response to Ms McDonald declining to rule herself in or out of the contest, before saying the candidacy decisions are "still at a very early stage".
Meanwhile, Independent Ireland leader Michael Collins has confirmed his party has spoken to a number of potential candidates about supporting any future presidential bid.
Mr Collins said "both" when asked if these individuals are in or out of politics currently, and declined to confirm a number of rumoured candidate names when asked.
He said that his party has told any candidate who has spoken with them that the party has four Oireachtas votes and that unless they can obtain 16 more votes from other sources, their presidential candidacy would be unlikely.
Mr Collins said Independent Ireland believes there is "no point talking to dreamers, there's a lot of dreamers out there".
President Michael D Higgins will leave office on 8 November next and the election to replace him is expected to take place late in October.
So far, no political party or group has formally selected a candidate.
