The Green Party is to enter formal talks with Fianna Fáil and Fine Gael, with a view to developing a programme for government.
Its Parliamentary Party made the decision after a meeting on Sunday.
Both Fine Gael and Fianna Fáil have agreed to enter government together - but need a third party to make up numbers in the Dáil.
During the week The Green Party set out its demands - however a key target of reducing carbon emissions by 7% was not agreed in a response.
On Friday, Taoiseach Leo Varadkar said it was something he would like to work towards.
Mr Varadkar has also said he is hopeful a new government can be in place by June.
In a statement on Sunday, The Green Party said: "We are conscious of the huge challenges facing any Government in the COVID-19 crisis.
"The party will now work with Fianna Fáil and Fine Gael to develop a deal that respects our mandate with a view to presenting that agreement to Green Party members for approval."
It added: "Green Party approval of any programme for government will require support of 2/3 of the Green Party voting membership.
"Any proposal must be transformative on climate action and commit to strong progress towards a more sustainable and fairer society.
"If this is not the case Green Party representatives will withdraw from negotiations and pursue their mandate in opposition and work to hold the government to account."
Green Party leader Eamon Ryan says he will seek a meeting with the Fianna Fáil and Fine Gael leadership to start the process.
An Taoiseach Leo Varadkar has welcomed the statement from the Greens.
He said: "We look forward to constructive discussions on a programme for government commencing in the coming days.
"This is a very welcome decision by the Green Party.
"Ireland needs a stable government to manage the remainder of the COVID emergency and to rebuild and renew our society and economy over the next five years.
"Working together we can help to deal with the challenges facing Ireland, including getting people back to work, re-opening businesses, investing in and transforming our health service, housing system, childcare, climate action, balanced regional development and enlivening communities in rural Ireland.
"We are also very aware that a hard Brexit within six months is a distinct possibility with all the challenges that would bring.
And Fianna Fáil leader Micheál Martin said: "As the country faces into a very uncertain future, it is critically important that we have a solid and stable Government in place.
"I am confident that working with Fine Gael and the Green Party we will be able to negotiate and agree a programme for government that protects people and ensures that the country’s economic recovery after COVID-19 is commenced as soon as possible and built on fairness.
"There is important legislation that needs to be passed to allow Irish businesses to access finance and there needs to be a government in place to do this.
"I am also confident that together, with an agreed progressive programme for government we will have the opportunity to tackle, in a fundamental way, the biggest issues that are facing our people - COVID-19 ,the housing emergency, access to health services, child care and climate change.
"We must also not forget that a hard Brexit in six months is still looming."
He added that Fianna Fáil "will be approaching the forthcoming programme for government negotiations constructively and confidently, determined to deliver on the mandate we have been given."
