Prime Minister David Cameron has spoken to Fine Gael leader Enda Kenny and congratulated him on his victory in the Irish general election.
A Downing Street spokesperson said Mr Cameron had invited Mr Kenny to Number 10.
A Fine Gael spokesman said Ireland's 85bn Euro EU/IMF bail-out was discussed in the conversation. Mr Kenny wants to re-negotiate the terms of the deal.
He has received calls from around Europe in the past 24 hours.
A Fine Gael spokesman said: "During all conversations, issues being discussed at European level in regard to the EU/IMF bailout were addressed in broad terms.
"Enda Kenny will be in attendance at a European People's Party Summit in Helsinki this coming Friday where he will meet many of the Leaders with whom he had conversations today."
European Commission President Jose Manuel Barroso said on Sunday he hoped for "constructive co-operation" with the new Irish prime minister.
Mr Kenny's party is now the largest in the 31st Irish parliament, the Dail.
However, they did not win an overall majority, and are expected to go into coalition with the Labour Party.
Negotiations about the coalition are due to start on Monday.
Counting is continuing in four remaining constituencies, with 13 seats in the new Dail (parliament) still to be filled.
A recount in Wicklow has been adjourned until Monday.
Meanwhile, Sinn Fein has won 13 seats so far, a gain of eight on its representation in the last Dail.
Fine Gael, Labour and Sinn Fein have all won record numbers of seats but FG's rivals Fianna Fail suffered a crushing defeat. Mr Kenny said it was "a democratic revolution at the ballot box".
Mr Kenny plans to start fighting for a cheaper loan deal on 4 March when the European People's Party, to which Fine Gael is affiliated, meets in Helsinki. He will follow that up at the European Council in Brussels the following week
Mr Kenny said the IMF/EU bail-out was "a bad deal for Ireland and a bad deal for Europe".
"We are not going to cry the poor mouth, other than to say the reality of this challenge is too much. I don't want to talk about difficulties, I look for co-operation, consensus and support across Europe," he said.
"We want to restore our pride at home and abroad. Our country is back in business."
The incoming taoiseach also pledged to probe deep into the heart of the Irish banking crisis which has left the taxpayer saddled with crippling debts which some analysts believe could ultimately cost closer to 200bn euros.
Fianna Fail leader Micheal Martin put on a brave face on his party's electoral collapse and said he believed that Fine Gael had managed to secure support from floating voters.
"There's a soft vote there for Fine Gael and Labour, just as there was for us for years," he said.
The party's biggest casualty was Mary Coughlan, the outgoing deputy prime minister, who was punished by voters in Donegal South-West as they opted to support an independent candidate in her own backyard.
"It's been a very difficult day for all of my colleagues, many of whom have lost their seats," she said.
Greens wiped out
The Greens - former coalition partners - were wiped out. None of their TDs - two of whom held Cabinet posts up until a few weeks ago - were brought back to the Dail.
Labour had one of its best ever performances with the prospect of supporting a coalition.
"That is the most likely outcome, there's no doubt about that," leader Eamon Gilmore said.
Sinn Fein has more than doubled its seats in the Dail with party president Gerry Adams topping the poll in Louth after giving up his Westminster seat for West Belfast.
Mr Adams said his party was on course for major gains: "I think the votes across the state show a significant amount of people support the position we have taken up."
Their success in the Republic of Ireland was extended with seats right around the border including two TDs in Donegal.
Further left wing appeal came from the United Left Alliance - a loose collection of the Socialist Party and the self-styled People Before Profit.
It will take two days to complete the official count.
Many parts of the country saw a big increase in turnout on the 67% recorded in the last general election in 2007.