While officials welcomed the calm start to negotiations on Monday, EU diplomats are expressing pessimism that Britain will agree a deal before it leaves the bloc.
At the same time, while it was a priority, he said, to show progress in averting difficulties for troubled Northern Ireland as a result of being on a new EU-UK border, agreement on how the border would work would depend on wider future trade ties.
The Union wanted a "fair deal", he said, renewing a vow not to punish Britain or take "revenge" for a withdrawal that has rocked the bloc and poses huge challenges for negotiators.
Davis, a veteran campaigner against European Union membership, said he sought quick and substantive progress in what is scheduled to be a two-year negotiation before Britain leaves the EU.
Chancellor of the Exchequer Philip Hammond said on Friday that protecting jobs and the British economy would be priorities for his country. The country is set to leave the European Union by the end of March 2019.
It will be their first meeting since her Conservative party unexpectedly lost its majority in a June 8 election, leaving her in charge of a so-called "zombie government".
Still, British Foreign Secretary Boris Johnson remained upbeat Monday, saying he thinks the Brexit negotiations will yield "a happy resolution that can be done with profit and honor for both sides".
"In a first step, we will deal with the most pressing issues".
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"Through exit negotiations, the United Kingdom government must ensure the United Kingdom continues to welcome, with minimal barriers, talented European Union students and staff".
"The four freedoms that give us the internal market must not be jeopardised", she said with reference to the EU's freedoms of movement of goods, capital, people, and services.
"It was clear from the opening that both of us want to achieve the best possible outcome and the strongest possible partnership".
"They should be agreed alongside each other".
"I am not in a frame of mind to make concessions, or ask for concessions", he said.
As recently as last week the government had insisted that trade talks could take place at the same time as discussion of citizens' rights, the Northern Irish border, and financial settlements.
According to Bloomberg, Soros said: "We are fast approaching the tipping point that characterises all unsustainable economic developments".
Central to this is the right to bring a non-EU spouse to Britain, which, when she was Home Secretary, Mrs May made more hard for Britons.
Mr Davis and Mr Barnier, both keen hikers, exchanged mountaineering-related gifts to mark the start of talks, with the EU's chief negotiator joking about the "steep and rocky path" ahead.