Social Democrats TD for Dublin Central Gary Gannon has ruled himself out for running for leadership of the party.
Party colleagues Holly Cairns, Jennifer Whitmore and Cian O'Callaghan have said they would take time to consider a leadership challenge.
Róisín Shortall and Catherine Murphy announced yesterday that they were standing down as party co-leaders.
Nominations for party leadership will open at 9am tomorrow and close at 12pm on Wednesday, March 1st.
The rules for the leadership election were agreed at a meeting of the party's National Executive this evening.
If there is a contested election, polls will close at the end of March and voting will be by digital ballot.
Anyone who has been a member of the party continuously for a period of no less than six months prior to the calling of the election can vote.
Voting will be by one member, one vote.
If there is only one candidate, that person will be deemed elected when nominations close on Wednesday, 1 March.
Speaking on RTÉ's Today with Claire Byrne, Mr Gannon said there were a number of ways to show leadership and he wants to be part of the team collective.
He said he believes his skills would be best placed in building the organisation.
Mr Gannon said Ms Cairns is "absolutely fabulous" and is a generational politician who appeals to a multitude of voters.
However, he added that the same could be said of Mr O'Callaghan and Ms Whitmore and he is "really excited about the next phase".
Ms Cairns, the TD for Cork South, has said she is going to take some time "this week" to consider if she will run for the leadership of the party.
She told RTÉ News: "I will be taking some time this week to consider and discuss all future leadership options with my colleagues, my family and my team."
Ms Whitmore told RTÉ's Morning Ireland that yesterday's announcement by "came out of the blue".
The Wicklow TD said that she was taking time to consider if she would run for the leadership of her party.
"It was a big day yesterday. Big day of emotions and at the moment, now I'm just going to take some time to think it through. I need to talk to my family and to my team, so I haven't made a decision, as of yet."
Mr O'Callaghan said yesterday that he had not made a decision on whether or not he will put himself forward for party leadership.
Mr O'Callaghan said yesterday that he had not made a decision on whether or not he will put himself forward for party leadership.
"I’ll be talking to supporters in my constituency over the next few days and taking soundings on it. So not ruling myself in or out at this stage," he told RTÉ’s Drivetime programme.
"Holly [Cairns] would be a fantastic leader, I think we’re in a lucky situation where we’ve a number of TDs I think that would be able to come forward and be good leaders of the party."
Ms Whitmore said that yesterday demonstrated that all her party members "work really well together as a team".
"Whatever happens and whoever goes for it, that will continue. We all have very different strengths. And I think all those strengths work very well together for the party," she added.
Ms Whitmore said there will be discussions about the co-leadership structure to see what is the best way forward for the party.
She said her "head is so full of all the permutations about the possibilities at the moment" and she rejected the suggestion that there was no need for two small social democratic parties in the Irish political system.
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She said that the last thing needed in politics was less choice. She said that a merger with the Labour Party was "not something that really features in our discussions".
"We are very, very focused on growing our own party, very focused on increasing our branches across the country, increasing our councillors and moving towards the local elections and the general election and that is our focus.
"So, whilst there can be a bit of discussion in the media in relation to this merger, it's not something that really features in our discussions at all."
In relation to possibly going into government with Sinn Féin, she said that "what's very important to us is our policies and being able to work on our policies and actually affect our policies", and any agreement on government would be decided by policies and getting social democracy ideologies into government decisions.
Additional reporting Paul Cunningham
