But even as outgoing Taoiseach Kenny directly warned Prime Minister May of her obligation to treat all of Northern Ireland's parties even-handedly, his successor, Leo Varadkar, held out the hope that the DUP's hold over Mrs May's majority might now mean "a soft Brexit" compared to her earlier stance.
Dodds will meet May at Downing Street on Thursday afternoon for one of a series of meetings May is having with the leaders of Northern Ireland parties in an effort to reach agreement on setting up a new power-sharing government for the region, he said.
But with so much at stake for Britain and its $2.5 trillion economy, pressure was mounting on May from within and without her party to heed other voices.
The Tories and the DUP are considering a "confidence and supply" arrangement which would see the Northern Irish party back the Government to get its Budget through and on confidence motions.
Before the election, May proposed a clean break from the European Union, involving withdrawal from the European Union single market, limits on immigration and a bespoke customs deal with the EU.
The DUP's main demands centre around securing more funding for the province, a source close to the party told Reuters on Saturday and Adams said he would welcome such funds.
Mr Verhofstadt told MEPs that while the door was open for Britain to change its mind on Brexit, there would be a price - set...
The newly-appointed Environment Secretary said the European Union referendum result must be honoured but it is "really important" that the exit package is "in the interests of the whole country".
If that's not possible, the Labour Party stands ready to offer strong and stable leadership in the national interest.
She declined to speculate about any change in Britain's plans to withdraw from the European Union following the election, saying, "The citizens have decided, and I assume that we will have to carry out these negotiations". Divisions over Europe helped sink the premierships of Margaret Thatcher, Major and Cameron, and many of her lawmakers and party membership support a sharp break with the EU.
The performance of the British economy could also influence perceptions: Data on Wednesday showed average weekly pay in the three months to April was down 0.4 percent, year-on-year, in inflation-adjusted terms - the biggest fall since the three months to September 2014.
Ahead of the start of talks Michael Gove, agriculture and environment minster, said aligning with Arlene Foster's DUP was aimed at "strengthening" the UK.
"I am concerned about the deal".
Major, who helped lay the foundations of the 1998 agreement that ended two decades of sectarian conflict in Northern Ireland when he was in government, said it would imperil the United Kingdom government's impartial role in the peace process.
"So there is a lot of anxiety and a lot of fear out of what is being discussed".
Any hard border, he said, would be catastrophic.