After the Conservatives lost their majority in the House of Commons in the 8 June general election, prompting calls for May to soften her line on Brexit, Davis made clear on Monday that the United Kingdom would leave the EU, but also the single market and the custom unions - the so-called hard Brexit.
However, Mr Davis said a deal on Ireland would be linked to a future trade deal and customs arrangements.
Johnson urged Europeans to look at the more distant future.
Still, British Foreign Secretary Boris Johnson remained upbeat on Monday and thinks that the Brexit negotiations will yield "a happy resolution that can be done with profit and honor for both sides".
N. Ireland is expected to be dealt with early on, between Oliver Robbins, UK prime minister's Sherpa, her chief diplomatic advisor and number two in the negotiations after Davis from the UK side and European Commission's Deputy Chief Negotiator Sabine Weyand on the EU-27 side.
Both EU negotiator Michel Barnier and his British counterpart David Davis said after the first negotiating session they were confident of quick progress but said major challenges lay ahead to meet the deadline of March 2019 for Britain to officially leave the bloc.
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May, who has faced criticism of her response to the tragedy , met victims, volunteers and community leaders at Downing Street. Cundy added that it will take weeks or longer to recover and identify all the dead in the housing block that was devastated.
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Subjects for the negotiations include the status of expats, the UK's divorce bill and the Northern Ireland border. On the opposite side is Michel Barnier, the French former foreign minister who is the European Commission's chief negotiator. In any case, European Union officials say, London no longer seems sure of what trade arrangements it will ask for.
Anxious by immigration and loss of sovereignty, Britainvoted a year ago to end its four-decades-old membership of the 28-country bloc - the first state ever to do so - in a shock referendum result.
Time running outAn increasingly concerned European Union has been pushing London to hurry up, with time running out for a deal and three months already passed since May triggered the two-year Article 50 European Union exit process. It was a clear rebuff to Mrs May's stated ambition of wrapping up a new free trade agreement quickly.
Today's talks are likely to focus on the practical details of timings for the coming months, with the big, divisive issues left aside for now, officials said.
Many in Brussels fear that London has no real strategy, with May under pressure at home, 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.
European stocks rose on Monday, partly on optimism about the talks actually getting underway after months of sniping and uncertainty, analysts said. The move backfired, May lost her Conservative majority in the vote and has been fending off critics of her leadership ever since.
Finance minister Philip Hammond confirmed Sunday that it was still the plan to quit not only the EU but the customs union and the bloc's single market as well.
"It's a statement of common sense that if we are going to radically change the way we work together, we need to get there via a slope, not a cliff edge".