However, a spokesman for Davis's Brexit department stressed that the UK's position had remained the same: "We have been crystal clear about our approach to these negotiations", said the spokesman. She tweeted that she felt that the DUP had leverage over the government.
Sinn Féin said any deal between the DUP and the Conservatives must not undermine the Good Friday Agreement.
One EU diplomat told the Politico website: "It is indeed our understanding that the agenda of the first negotiation round consists of issues related to the first phase of negotiations, which means citizens, money, Northern Ireland and some other exit-related questions".
Declan Kearney, the party's chairman, said a DUP focus on Ms Foster's future role at Stormont "is completely misdirected and premature".
The Sinn Fein leader added that he and his colleagues handed over the late Martin McGuinness's resignation letter, written when he stood down as Northern Ireland's deputy first minister in January.
"However, we need to do it in a way that respects the specific circumstances of Northern Ireland, and, of course, our shared history and geography with the Republic of Ireland".
Mr Adams said they would be pushing Mr Varadkar to raise the prospect of a border poll on Irish reunification within the next five years in discussions with the British government.More news: 2 inmates remain on run after 2 prison guards killed
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Ms Foster dismissed suggestions her deal with the Tories threatened the peace process.
"I will be speaking to her about a renewed commitment by the two governments to work together to ensure that the (Northern Ireland) executive is established before June 29", Leo Varadkar told national broadcaster RTE on Friday, in reference to the deadline to restore the power-sharing regional government. "What would happen then?"
And one in five businesses said that they would target future growth outside the United Kingdom or even relocate as a result of the Brexit vote.
Grainne Teggart, Campaign Manager for Amnesty International in Northern Ireland, said: "Today's results confirm what we have long known: that an overwhelming majority favour reform of our inhumane abortion laws".
He said: "We're Irish republicans, we have just been elected on a certain mandate".
However, it also removes a critical vote that could have toppled the Government and comes as a crucial support deal with the Democratic Unionist Party has yet to be finalised.
But some opposition politicians say that May can no longer stick to her stance for a clean break with the European Union, characterising her election bid as a poor gamble that has left Britain a laughing stock.