The European Commission released a statement at the time saying discussions would focus on "issues related to citizens' rights, the financial settlement, the Northern Irish border and other separation issues, as part of the sequenced approach to the talks".
Mr Davis said Britain's negotiating position had not changed as a result of his Conservative Party's poor showing in recent elections.
President Trump, who has increasingly shaky relationships with European leaders, was a vocal supporter of the United Kingdom divorcing from the E.U. during the presidential campaign.
The EU says Britain can't leave without settling its bill, paying up for all its commitments that are still ongoing, including projects that might reach into the next decade, as well as the U.K.'s share of EU staff pensions.
Mr Murray said it was "pie in the sky" to suggest that a trade and customs deal would secure the same access for to the European single market as EU membership.
He vowed to seek "a deal that works in the best interests of all citizens" with Mr Barnier as the pair began their discussions at the commission's Berlaymont headquarters in the Belgian capital.
Davis said there was a "long road ahead" but predicted a "deep and special partnership".
Given that a disorderly departure would have greater impact on the United Kingdom economy than Europe's, officials have warned that Brussels would have the upper hand in several aspects of the talks because of the limited time available to conclude a wide-ranging exit deal.
Mr Davis said Prime Minister Theresa May would brief fellow European Union leaders at a summit on Thursday on the UK's approach to the rights of expatriate citizens, which will be set out in detail in a paper on Monday.
"We want both sides to emerge strong and prosperous, capable of projecting our shared European values, leading in the world, and demonstrating our resolve to protect the security of our citizens".
"There is more that unites us than divides us", Davis said, referring to the latest reported terror attack overnight in London and the loss of lives in forest fires in Portugal.
Both sides were determined to give the talks a positive spin, with Mr Davis and Mr Barnier exchanging mountaineering gifts - a walking stick and a book on hiking - ahead of the all-day meeting.
When Theresa May sent the EU Britain's formal notice to quit the EU in March, she wrote: "We therefore believe it is necessary to agree the terms of our future partnership alongside those of our withdrawal from the European Union".
However, EU leaders are also said to be determined not to allow concessions to Britain that might encourage others to quit.
"It's been a year since the referendum".
Chancellor Philip Hammond has dismissed speculation that he will push for a soft Brexit by arguing for Britain to remain in the customs union and single market.
May officially triggered the two-year Brexit process in March when she was riding high in the opinion polls.