"It was clear from the opening that both of us want to achieve the best possible outcome and the strongest possible partnership - one that works for the United Kingdom and the EU".
He and others have also raised the possibility that Britain could remain in the EU.
UK Brexit Secretary David Davis said that the Brexit talks had gotten off to a "promising start" even though it has become obvious that Britain conceded to the EU's preferred order.
The EU's chief negotiator, Michel Barnier said: "We are talking about orderly withdrawal first and that makes sense".
Given the floor for 10 minutes at the end of a Brussels summit dinner, her first since she launched the two-year withdrawal process in March, May outlined five principles, notably that no European Union citizen resident in Britain at a cut-off date would be deported.
Davis has said he was "encouraged by the constructive approach" as the talks began.
Speaking to an audience packed with banking executives, he also took a swipe at moves by the European Union to try to shift some of Britain's financial services to the bloc.
In their June 2016 referendum, the majority of the British citizens voted in favor of Brexit (51.9% to 48.1%). We must lift the uncertainty caused by Brexit.
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He said: "The position hasn't changed, we have the Lancaster House speech, the two white papers, and the Article 50 letter, all backed up by a manifesto too".
The UK has accepted that the European Council will decide when "sufficient progress" has been made on these three baskets to allow talks on the EU/UK future relationship to begin.
She added: "We want to conduct these negotiations in such a way that they take place in a good spirit, and we know that we want to continue working afterward with Britain".
The most senior officials on either side will lead work on efforts to resolve the problem of the border between Northern Ireland and the Republic of Ireland, a situation Mr Barnier acknowledged was "politically sensitive" at a time when the Tory government was seeking the support of the Democratic Unionist Party to prop up Mrs May's minority government.
Ian Murray, the Labour MP for Edinburgh South, called on Mr Davis to abandon any suggestion that the United Kingdom could walk away from Brexit talks without a deal.
The Sun, which backed May's center-right Conservatives in a June 8 election that cost her her parliamentary majority, also noted Davis had gone along with Brussels' plans after saying he would make getting immediate trade talks "the fight of the summer".
The two chief negotiators met in Brussels and, after an intricate discussion, they both expressed their confidence regarding the separation of Britain from the European Union. More recently, however, it appears to have renegaded on these aims, with Prime Minister Theresa May insisting that Britain should be willing to relinquish any partnership with the European Union if a suitable deal can not be struck within the two-year negotiating time frame.
Asked about the timeline of Brexit negotiations, Mr Varadkar said: "I think what's much more important than the timetable is the outcome". "It's not about punishment, it's not about revenge". "The protection of the Good Friday agreement and the maintenance of the Common Travel Area are the most urgent issues to discuss", the European Union negotiator explained. This uncompromising stance is "not about punishment" or "revenge", but simply a effect of Britain's decision to exit, he said.
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