The Green deputy leader Catherine Martin has said she will give "serious consideration" to contesting the party's forthcoming leadership election.
However, she said in a statement this evening that her "focus remains on the crucial government formation talks".
Deputy Martin added: "I believe it is important that government formation talks fully conclude uninterrupted and that any leadership campaign happens subsequently."
Four Green Party councillors have written to Ms Martin "urging" her to challenge Eamon Ryan for the leadership of the party.
Cork City and County councillors Lorna Bogue, Oliver Moran, Colette Finn and Liam Quaide wrote to Ms Martin yesterday evening, saying they believe she and not Mr Ryan is "prepared to make difficult choices for the greater good".
In the letter, seen by RTÉ News, the councillors state: "In February, Ireland voted for change. We believe with your style of leadership, your convictions and your work ethic, you are the right person to lead the Green Party.
"Regardless of the outcome of these [government formation] talks, we will need a leader who we trust to do the right thing for the country and the party.
"Whether we enter government or not, we believe the party needs a new leader, someone who will fight hard for all of our futures.
"We urge you to put yourself forward as a candidate for the leadership of the Green Party. You have our support."
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The letter comes as the Green Party continues to hold government formation talks with Fine Gael and Fianna Fáil, after the inconclusive 8 February General Election.
The Green Party won an historic high of 12 Dáil seats almost a decade after losing all of its seats in the 2011 general election and four years after they won back two seats, those of Mr Ryan and then first-time TD Ms Martin.
However, while the increase in support has come under Mr Ryan's leadership, the decision to enter talks with Fine Gael and Fianna Fáil has led to divisions within the Green Party.
Under Green Party rules, a vote to decide on who will be its leader must take place within six months of any general election, with nominations for any potential contest opening on Tuesday.
The councillors' letter is likely to add to fresh focus on the government formation talks over the coming weeks.
This is in part because the Green Party's negotiations team is led by Ms Martin and any potential decision not to ultimately enter government could lead to fresh questions over Mr Ryan's leadership.
The Green Party's 12 TDs voted by eight to four in favour of entering government formation talks at a private parliamentary party meeting late last month, with Ms Martin voting against the proposal.
Mr Moran said he would like Ms Martin challenge for the leadership of the party because it needs to broaden its appeal and she resonates strongly with new voters on issues such as health, housing and education.
He said Ms Martin, who is from Co Monaghan, also "speaks the language of rural Ireland" and is a very strong candidate.
Speaking on RTÉ's Morning Ireland, he said Mr Ryan has done a fantastic job as party leader but there is a groundswell for change.
Mr Moran said he has spoken to Ms Martin and hopes she will decide to go forward so members of the party can have a say in who the next leader should be.
The @greenparty_ie MEP @CiaranCuffe re leadership election says: "Not inclined to comment, but happy enough that some members believe we don't subscribe to the 'una duce, una voce' school of politics." @rtenews— Paul Cunningham (@RTENewsPaulC) May 20, 2020
Green Party TD for Dublin South West Francis Noel Duffy said a leadership election "is part of our constitution" and therefore the party is "tied into it."
Referencing the government formation talks, he said he believed "all measures have been taken to hold it outside of the current process".
However, he added, "time has dictated where we find ourselves".
Mr Duffy is married to Ms Martin.
Several Green TDs and Senators have today come out in support of Mr Ryan remaining as leader.
"He has led us through the very low times and grown us to a party of 49 councillors, two MEPs and 14-member Parliamentary Party," said TD for Wicklow Steven Matthews.
"That type of growth will always generate healthy competition and ambition in others. If Eamon Ryan wishes to continue as leader he will have my full support.
"In my experience there is nobody comes close to him in terms of leadership qualities, fairness and vision and I will continue to support him".
Green TD for Dublin West Roderic O'Gorman said he "looks forward to nominating Eamon Ryan" for leader, if there is a contest and that Mr Ryan will have his "Number 1" in any election.
Green TD for Carlow Kilkenny Malcolm Noonan has also strongly backed Mr Ryan.
He said: "The [Government formation] talks are going well and our sole focus should be on being in a position to present a transformative draft programme for government to our members and providing stable government to help people through this crisis."
He added: "Eamon has led our party from virtual wipeout to our greatest ever electoral success. He has my full and unwavering support should he wish to stay on as party leader.
"We need to honour the mandate we've received by implementing the policies on climate and biodiversity that we have been talking about for far too long. That should be our only task right now."
Green TD for Waterford Marc Ó Cathasaigh said talk of a leadership election in the party is an "unwelcome distraction from the important job of work we have in front of us at the moment".
He said: "The Green Party are in programme for government talks that have the potential to deliver positive, transformative change in Irish society over the coming decade. All our energies should be focused on delivering the best possible outcome for our voters and future generations."
Supporting his leader, he added: "Eamon has led us from our lowest ebb to our best ever electoral results in both local and national elections.
"I am confident of his ability to lead us through this negotiation process and deliver a programme for government that will be accepted by our membership. Now is not an opportune time to consider changing leadership."
Green TD for Dún Laoghaire Ossian Smyth has also backed Mr Ryan's leadership.
?He said: "I back him 100% and I ask any other contenders for the leadership to wait until the convention in October before mounting a challenge. Holding a contested leadership election right now would be a huge distraction at a critical time for the country and for the environment."
"In the past year, Eamon has led the party to a quadrupling of our membership and a quadrupling of our elected representatives. One of Eamon's great strengths has been his ability to recruit a range of gifted, electable candidates."
Green Party Senator Pippa Hackett said she would "see a contested election at this time as totally unnecessary".
She added: "Under Eamon's leadership this party has grown, not only in numbers, but geographically, and that has been a hugely important development for us. I have every confidence in Eamon now, and into the future."
Ms Hackett's Seanad colleague, Senator Pauline O'Reilly, added: "Eamon has my full confidence and support as leader of the party, now and in the future. Our focus is on government formation talks and we won't be distracted."
Meanwhile, Green MEP Ciaran Cuffe has said he was "not inclined to comment" regarding calls for Ms Martin to contest the party's leadership election but adds "... happy enough that some members believe we don't subscribe to the 'una duce, una voce' school of politics".
Additional reporting Fiachra O'Cionnaith