The Fianna Fáil TD in Cork North-Central said the party has to be open to negotiating with any party if it is serious about developing a programme for government.
However, the Solidarity TD in the area has urged Sinn Féin and the Greens not to "rescue" Fianna Fáil or Fine Gael after the election.
Fianna Fáil's Padraig O'Sullivan, speaking after he retained his seat in the constituency, said that the vote sent a clear message that the public do not want another confidence and supply arrangement.
Mr O'Sullivan declined to be drawn on whether he would be happy to work with Sinn Fein, instead insisting that it is better to wait to "see how the numbers fall" instead of speculating on future governments, though he did say that all options should be considered.
"Nationally, the trend hasn't been as kind to us but, at the same time, we are happy with the work we did here," he said.
"There are a number of permutations between forming coalitions or confidence and supply with various parties; all avenues need to be explored. We do need to open to speaking to everybody. You have to be able to agree a programme for government, to agree a deal.
"There is no point in us pretending that another confidence and supply would be to everyone's satisfaction."
Solidarity TD Mick Barry urged those parties on the left not to back a Fianna Fáil or Fine Gael government, though.
He was speaking after retaining his own seat in the constituency.
"The election has shown a desire among ordinary people for change," he said.
"The big beneficiaries in this election are Sinn Féin and, around the country to an extent, the Greens.
Those parties should now rule out doing deals with Fianna Fáil or Fine Gael. I don't think those parties should be rescued.
The other big winner on day two of the count was Colm Burke, the Fine Gael Senator who finally won a seat after nearly four decades of trying.
The Tánaiste, Simon Coveney, was the first to pay tribute to Mr Burke.
"I think he deserves it. He is an incredibly hard-working Oireachtas member," Mr Coveney said.
"People respect the work-rate of Colm Burke and they know he will represent people on the ground from all sorts of backgrounds."
Mr Burke faced challenges including, most notably, the internal issues in Fine Gael in Cork North-Central, where Dara Murphy's representation had come under heavy scrutiny in the wake of questions about his attendance and expenses.
Mr Burke said, "I think the great thing for Cork North-Central, for Fine Gael, it is a huge boost. We have had a difficult time over the last few years and it is important we have a strong representative for those people in the constituency."
For others, it was time to reflect on results.
The Greens played down suggestions they would be disappointed by failing to take a seat in Cork. Just last week, party leader Eamon Ryan targeted 15 seats, including four in Munster.
Oliver Moran, the Cork North-Central candidate, said the party is proud of its performance.
He said, "It shows how far we have come. It is not a green wave, it is a desire for change.
"To pull off four in Munster, we needed two in Cork but the performance has been strong. We have proven that our mandate from the local elections was not a flash in the pan. We are consolidating on a loyal base."
Labour, too, found themselves on the outside looking in.
John Maher, who had been tipped by many as an outside bet for a seat, was eliminated early on and described himself as "gutted".
He said the new TDs have a big job to deliver for Cork North-Central.
"Enough of the noise and shouting from the ditches, people must take up the responsibility of being elected," he said.