Theresa May is expected to provide her fellow European Union leaders with detail on the status and rights which the United Kingdom plans to offer EU citizens living in Britain after Brexit.
Mr Barnier spoke at a press conference after the first day to confirm that settling priority issues would build trust on which to negotiate a new relationship.
He said "we are witnessing the return of the European Union rather as a solution, not a problem".
Britain's Brexit minister, David Davis, said he was "encouraged" by talks at the first meeting between British and EU Brexit negotiators on Monday.
THERE WAS no escaping the symbolism of the gifts exchanged by Brexit Secretary David Davis and European Union chief negotiator Michel Barnier at the start of the talks.
Britain is "very conscious of how they will use that time sequence to pressure us, and we'll avoid that at every turn", Davis told ITV on May 14.
Most of the attention will be paid to the negotiations on the future relationship with Ireland.
Attending her first European Council Summit since her disastrous general election, it is understood the Prime Minister will explain the UK's "principles" on the issue of citizens' rights - a key early topic in the Brexit negotiations.
Davis has said he was "encouraged by the constructive approach" as the talks began.
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The U.K. and the European Union started talks on Britain's exit from the bloc Monday morning, nearly a year after the U.K. voted to leave, with EU chief negotiator saying he hoped the two sides can start removing the uncertainties created by that decision.
The UK will publish a detailed paper on Monday "outlining an offer" on citizens' rights, Mr Davis said, which he hoped would lead to a swift agreement.
Opponents of Brexit seized on the EU's success in imposing its timetable for the talks as evidence of the UK's weak negotiating position.
Krzysztof Lapinski reacted Thursday to criticism by Macron who said in an interview with newspapers including Poland's Gazeta Wyborcza daily that East European leaders are betraying Europe's values but want its funds, adding: "Europe is not a supermarket".
Regarding the future relationship with the European Union, the secretary said that Britain hasn't changed its position despite the Conservative majority was wiped out after the snap election on June 8. We want to make sure that the withdrawal of the United Kingdom happens in an orderly manner.
Mr. Barnier has said he hopes to conclude the talks by October 2018, leaving six months to ratify the deal.
The two sides agreed to set up three working groups covering the details of withdrawal: the financial settlement; citizens' rights; and one dealing with other legal issues relating to "separation".
European Union diplomats hope this first meeting, and a Brussels summit on Thursday and Friday where Theresa May will meet fellow European Union leaders, will improve the mood after some spiky exchanges.
Britain and the European Union finally began formal Brexit negotiations Monday, vowing to work constructively for a deal despite disarray in London over whether to go for a "hard" or "soft" divorce. Michel insisted the British people had chosen to leave and negotiations should now proceed apace.
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