The Sinn Féin leader Mary McDonald has said she will be meeting with other party leaders this week and hopes to meet with Fianna Fáil leader Micheál Martin.
Speaking on RTÉ's This Week programme, Ms McDonald said while she had campaigned for a government without either Fianna Fáil or Fine Gael, the worst possible outcome would be a government with both.
She accused Mr Martin of 'looking down his nose' at the people who voted for Sinn Féin, saying Fianna Fáil has a decision to make as to whether it wants to put Fine Gael back into government.
"They're [Fianna Fáil] talking to Independents, they've talked to others. Interestingly, the only group that they've steadfastly refused to speak to is the second largest party in the Dáil, and I really think for Micheál Martin to look down his nose at over 400,000 people who voted for Sinn Féin and to disregard those votes en masse is really a bad look."
She said Sinn Féin in government would mean "a better outcome for lots of issues - housing, investment in communities, getting immigration right. The last government got it terribly, terribly wrong, in a way that was divisive".
Ms McDonald said she has already met with the Social Democrats and hopes to meet with Labour in the next week, adding that she also hopes to have the chance to meet with others, including Mr Martin.
"I think the least that he [Martin] can do is show a level of respect to the Sinn Féin electorate," she added.
Ms McDonald said the election result shows that Sinn Féin's success in 2020 was "not a flash in a pan" and "confirms Sinn Féin as a significant political force south of the border".
Sinn Féin finished second in the General Election with 39 seats in the Dáil behind Fianna Fáil with 48, but ahead of Fine Gael with 38.
Fianna Fáil and Fine Gael, with a combined 86 seats, are just short of the 88 required for a majority.
If they wish to return to government together, they would need one smaller party as a junior partner or a handful of independents.
Mr Martin has previously indicated he did not plan to go into government with Sinn Féin.
FF TD indicates preference for Fine Gael coalition with Independents
Fianna Fáil TD Jim O'Callaghan said he would prefer his party and Fine Gael combine with Independents to form a government.
Speaking on the same programme, he said that there is a group of "reliable Independents" which he thinks could be relied on to form a stable government.
He said the fact that Fianna Fáil won more seats in the election should be reflected in the composition of the government and his party should hold the office of Taoiseach for longer if the post is rotated as "that would have to be reflected by the proportionality of the strength of each party in the Dáil".
He added that the new government should be formed before US President-elect Donald Trump is inaugurated in Washington on 20 January.
Labour and the Social Democrats held talks about the post-election landscape at Leinster House on Friday.
The Labour Party had called for a centre-left platform to engage with the larger parties as part of talks to form a government.
However, Social Democrats deputy leader Cian O'Callaghan said they would be going it alone and plan to meet with Fianna Fáil and Fine Gael next week.
The Dáil is set to meet on 18 December however it is unclear if political talks will have lead to a new coalition government by then.
