She said she assumed that the talks would proceed despite the setback for Prime Minister Theresa May's Conservatives in the June 8 parliamentary election, in which her party lost its governing majority.
May called the early election in April when opinion polls suggested she was set for a sweeping win which would give her a strong mandate to start Brexit talks with the EU.
But her vulnerability at home means that her plan to leave the EU's lucrative single market and customs union in order to impose strict limits on immigration is under intense scrutiny.
'I think there is a unity of objective among people in the United Kingdom, ' May said following a meeting with French President Emmanuel Macron on Tuesday.
After a tumultuous week that pitched Britain into its deepest political crisis since the Brexit referendum a year ago, May's future was uncertain, darkened by her botched gamble on a snap election and muted response to a deadly fire in London.
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May said on Tuesday that talks with the DUP had been productive - a view shared by DUP leader Arlene Foster - and that Brexit negotiations would begin as planned next week.
"That is why we are pushing ahead with negotiations on Monday".
The United Kingdom general election in 2017 took place on Thursday 8 June 2017. That's why we're ready to start very quickly. "I can't negotiate with myself".
At a press conference with May, Macron said the door was "always open" for Britain to remain in the European Union as long as the negotiations on Brexit have not finished.
Barnier said: "All options are available and on the table".
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"It's imperative that both governments recommit to the word, spirit and implementation of the Good Friday Agreement if there is to be any prospect of re-establishing the Executive [in Northern Ireland]".
Under the proposed deal, the DUP would likely support May's Conservatives on big issues such as the budget, Brexit and defence legislation on a vote-by-vote-basis.
Sinn Fein and the Democratic Unionists both agree a deal to restore a powersharing administration in Northern Ireland can be done by the end of the month.
London's neutrality is key to the delicate balance of power in Northern Ireland, which was once plagued by violence over Britain's control of the province.
Besides forging a deal to keep her job and preparing for Brexit talks, May is grappling with a crisis over the breakdown in power-sharing between the pro-British DUP and the Catholic nationalist Sinn Fein party in Northern Ireland's separate, devolved government.
Hot, cloudy, muggy in the forecast for Father's Day weekend
More of the same tomorrow, with showers and thunderstorms developing in the afternoon and continuing into the early evening. As the front pushes through, the humidity will go up and temperatures will likely peak in the 70s to near 80 degrees.
Conservative Prime Minister Theresa May first triggered the Article 50 mechanism to leave the bloc nearly three months ago, with formal discussions due to begin on 19 June. Labour had won 261 seats.
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