The leader of Sinn Féin, Mary Lou McDonald, has said the party will be "intensifying" its discussions to form a government this week.
The party won the second most seats in the recent general election, just one seat behind Fianna Fáil.
The Fine Gael Parliamentary Party is meeting on Monday for the first time since finishing third in the election.
Acting Taoiseach Leo Varadkar has said the party is preparing for opposition - and will only enter government as a last resort.
Ms McDonald said: "Sinn Féin wants to form a government for change and this week we will be intensifying our discussions with the Green Party, Social Democrats, Solidarity/People Before Profit, Independents4Change and Independents to explore how we deliver a government which will build homes, cut rents and freeze them, end the trolley crisis, deal with climate change, give workers and families a break and advance Irish unity."
"Everyone agrees that last week the people voted for change. The question for all parties is are they willing to be part of a government for change?"
"It is clear at this stage that Fianna Fáil and Fine Gael want to block change and maintain their grip on power.
"This is not what people voted for. In fact they voted against the very policies implemented by Fine Gael and Fianna Fáíl over the last four years.
"That is why there was huge anger at the weekend at the prospect of Fianna Fáil/Fine Gael being returned to government, despite the outcome of the election."
While acting transport minister Shane Ross earlier said he believed a Fianna Fáil/Fine Gael coalition was "the only real option".
He said: "I think what we're now doing is a kind of a mating dance with a result - which looks like it's going to be a Fianna Fáil/Fine Gael coalition.
"They'll both say they're doing it incredibly reluctantly but they actually want to be in power, and they don't want to be in power with Sinn Féin.
"It's what they're avoiding more than what they're actually looking for.
"I think we will have a Fianna Fáil/Fine Gael coalition with The Greens in it.
"Nobody's going to call them together in the immediate future - we're going to have obviously a non-event in the Dáil on Thursday.
"And then they may include some Independents; and I notice some of the Independents are doing something similar to what we did last time - which was get together and say 'we're a group now, come and talk to us as well'".