Sinn Fein's Michelle O’Neill has said that she is ready to resume talks on Monday with the aim of restoring the Stormont government after a "watershed election".
The Northern Ireland leader said that nationalist opinion is l"ooking increasingly to leadership across Ireland rather than to Britain".
Ms O’Neill said: "This was another watershed election. For the first time since the foundation of the state the unionist vote was less than 50% in a Westminster election.
"This was also a vote against Brexit. It was significant election with the Sinn Fein vote increasing in all constituencies and electing seven abstentionist MPs.
"Nationalist opinion is looking to leadership at home and across Ireland not Britain.
"The current issue of a hung British parliament is transitory. It is no surprise that the DUP have sided with the Tories.
"Experience shows that unionists have minimal influence on any British government be that on Major, Thatcher or Theresa May.
"They have achieved little propping up Tory governments in the past and put their own interests before those of the people.
"Sinn Fein is entering the talks to re-establish an Executive that delivers for all and for the full implementation of the agreements.
"We need the parties acting together to secure special designated status within the EU for the North.”
Gerry Adams says Leo Varadkar "needs to stand up to the challenges as a co-equal guarantor of people's rights and of the institutions" pic.twitter.com/hUUTXnkL6X— RTÉ News (@rtenews) June 9, 2017
Sinn Fein president Gerry Adams said: "This was another momentous election. There is a need for reflection. History shows that British governments betray unionists."
Mr Adams also said that incoming Taoiseach Leo Varadkar "needs to stand up to the challenges as a co-equal guarantor of people's rights and of the institutions".
DUP press conferencePosted by Belfast Telegraph on Friday, June 9, 2017
Arelene Foster set for talks with May
Earlier DUP leader Arlene Foster has said she will enter talks with Theresa May to "explore how it may be possible to bring stability to our nation" to form a new government with the Conservative party.
Mrs Foster said: "I make no apology for saying that the DUP will always strive for the best deal for Northern Ireland and its people. But equally, we want the best for all of the United Kingdom."
Mrs May has said she will work with her "friends and allies" in the DUP to provide certainty for the UK in its forthcoming Brexit negotiations.
Theresa May 'sorry'
Meanwhile Theresa May has said she is "sorry" for Conservative MPs who lost their seats after her General Election gamble backfired.
The Prime Minister said the unseated MPs - including eight ministers - had not deserved to be ousted as she saw her Commons majority wiped out.
"I am sorry for those candidates and hard-working party workers who weren't successful but also particularly sorry for those colleagues who were MPs or ministers who had contributed so much to our country and who lost their seats and didn't deserve to lose their seats."
Mrs May acknowledged that she had gone into the election - which she did not have to call for another three years - hoping for a "large" majority.
"As I reflect on the result, I will reflect on what we need to do in the future to take the party forward," she said in a broadcast interview
Earlier, following an audience with the Queen, Mrs May said she would seek to lead a minority government supported by the Democratic Unionists (DUP).
"What the country needs more than ever is certainty, and having secured the largest number of votes and the greatest number of seats in the General Election, it is clear that only the Conservative and Unionist Party has the legitimacy and ability to provide that certainty by commanding a majority in the House of Commons," she said in a statement on the steps of No 10.
"As we do, we will continue to work with our friends and allies in the Democratic Unionist Party in particular.
"Our two parties have enjoyed a strong relationship over many years, and this gives me the confidence to believe that we will be able to work together in the interests of the whole United Kingdom."
George Osborne's Evening Standard sticks the boot in
Meanwhile former chancellor George Osborne's Evening Standard has stuck the boot into Mrs May, saying her "authority is non-existent"
A damning editorial piece published in Friday's paper states: "We now have a minority Conservative government that is in office but not in power.
"The DUP does not support some central tenets of the Government’s economic and welfare plans.
"In this topsy-turvy world, the decisions that affect London will now be taken in Belfast.
"She herself said: "If I lose just six seats, I will lose this election". Team May lost twice that number.
"As an unelected premier, she had every right to seek a mandate. But she failed to frame what the election was about."