Davis said the talks were off to a "promising start" and denied that Britain had caved in on the sequencing of the talks.
British Brexit Secretary David Davis said the country had not backed down over when talks could turn to a future trading relationship.
In their June 2016 referendum, the majority of the British citizens voted in favor of Brexit (51.9% to 48.1%).
Speaking after his first meeting with European Union chief Brexit negotiator Michel Barnier since being appointed to his new role, Mr Coveney said Ireland's unique circumstances are well understood in Brussels.
He said May will publish a detailed paper outlining Britain's approach on protecting citizen rights at the European Union summit on Thursday and Friday, which he believes "will form the right basis on which to reach agreement".
In Berlin, German Chancellor Angela Merkel emphasised the unity of the remaining 27 European Union countries, who have been alarmed in recent weeks by May's threats to walk out of the talks.
The UK agreed to settle the issue of citizens" rights, borders and the so-called "divorce bill' with the European Union before discussions over trade begin, when formal Brexit negotiations finally got under way in Brussels on Monday.
But the point was effectively conceded before talks began, with Mr Davis yesterday accepting an European Union timetable set out last week, which makes clear trade will only be discussed once "sufficient progress" is made on citizenship, Northern Ireland and a "single financial settlement" of as much as £88 billion.
Barnier, following the talks in a joint press conference with Davis, said: "Today we agreed on dates". And it was useful for me to sit down with my counterpart, David Davis.
U.S. coroner investigating death of student freed from N Korea
The 22-year-old Warmbier, who was released in a coma last week after almost 18 months in detention in North Korea , died Monday. He had severe brain damage and was sent from North Korea on 13 June to a hospital in his home city of Cincinnati where he died.
But many in Brussels fear that London has no real strategy, with May under pressure at home and still trying to close a deal with a conservative Northern Ireland party to stay in power, and facing criticism for her handling of the aftermath of a devastating tower block fire. We must lift the uncertainty caused by Brexit.
Mr Davis - who earlier said that he was hoping to negotiate a "deal like no other in history" - said that the United Kingdom was looking for "a new, deep and special partnership with the EU".
The UK had wanted the future of trade relations to be considered alongside other issues but the opening day of talks concluded with agreement that progress will be made on citizens" rights, the so-called "divorce bill' and border relations between the Republic of Ireland and Northern Ireland first. Perhaps more important, they begin just 10 days after a humbling election setback revived divisions within May's government on what kind of relationship it wants with the EU.
Davis insisted that talks about trade would occur "in parallel", but Barnier said they would start only in a second phase.
Davis said before the talks began that he is determined to reach a Brexit deal "that works for the whole of the United Kingdom".
The arrangement most like that seems to be Canada's, which does not honor free movement of European citizens nor pay money to Brussels, but which does now enjoy a free-trade agreement with the huge bloc.
The two sides agreed to initially hold a week of negotiations every month until a new structure is defined.
"It's not when it starts it's how it finishes that matters", he said. This uncompromising stance is "not about punishment" or "revenge", but simply a outcome of the U.K.'s decision to exit, he said. "It's not about punishment, it is not about revenge", Barnier said. An EU official said Britain still refused to accept it would owe Brussels anything.
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