Davis, a veteran Eurosceptic, told Barnier that his team aimed to maintain a "positive and constructive tone" during the talks.
Uncertainty over the future trading relationship also means that special status in the talks has been given to discussions on how to limit the impact on peace in Northern Ireland from the creation of a new EU-UK border with the Irish Republic. An EU official said Britain still refused to accept it would owe Brussels anything.
In a nod to the difficulty of that issue, Mr Davis said it took up more time "than anything else".
Roth said that "Brexit is a very, very hard operation" and there's only a bit over a year to negotiate it.
Mr Barnier said the meeting was a "useful" opportunity to "get off on the right foot", and quoting French diplomat and European Union founding father Jean Monnet, he said he was "neither optimistic nor pessimistic" but "determined".
Mr Barnier's insistence on sticking to the EU's priorities for the negotiations comes after Mr Davis appeared to soften his stance on the schedule for the talks. The EU treaties have set a timeline of two years for the exit, which means that at the end of March 2019, the United Kingdom will leave the EU.
Brexit Secretary David Davis will call for "a deal like no other in history" as he heads into talks with the EU.
Until now, British prime minister Theresa May had insisted that both discussions should be held in parallel.
As required by the European Union, negotiations will start with the divorce itself, and when "sufficient progress" has been achieved, they will move onto discussing the future relationship between the bloc and the UK.
Mr Davis said the two sides hoped they could reach an agreement of citizens' rights "reasonably fast".
May expects the process to be completed by March 2019 but the European Union has warned that the two sides should reach a deal before October 2018. EU nations and the European Parliament will have to approve any future deal and that can take months.
The letter, from groups representing hundreds of thousands of businesses, is the latest evidence of efforts to steer the Government away from the "hard Brexit" which they fear will harm the economy, see prices rise and jobs losts.
In fact, 53% of Britons would favour a second referendum on the UK's European Union membership, according to a Survation poll published by the Mail on Sunday; only 47% favoured a second referendum in April according to the same pollster.
"A fair deal is possible and far better than no deal", Barnier said at a press conference on Monday.
Quoting Winston Churchill, Davis said in response: "A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty".
"It was clear from the opening that both of us want to achieve the best possible outcome and the strongest possible partnership", Davis said.
The socialist Gabriel said the Conservative government in London "played with the emotions of citizens in Britain, told fake news about Europe and left people unclear about what consequences this would all have" and "they bet on a snap election because they thought they could use tactical tricks to get a bigger majority".
Egypt's Azhar condemns London mosque attack
To the north, Manchester was hit by a deadly attack May 22 when a suicide bomber killed 22 people at an Ariana Grande concert. It said searches were being carried out at a residential address in the Cardiff area in Wales .
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After realising that she had a snake in her hand instead of a toy, she ran away for her life throwing the snake on the ground. Ariel Rosso posted this video on YouTube of his wife Carla. "Check it off my bucket list".
Historic Brexit divorce negotiations begin in Brussels
Both sides acknowledged the clock was ticking, with the date for the UK's departure from the European Union fixed for March 2019. We also agreed on how we will structure our talks. "It's not when it starts it's how it finishes that matters", he said.
In a first, Ford to import Focus small car from China
Be proactive - Use the "Flag as Inappropriate" link at the upper right corner of each comment to let us know of abusive posts. Ford said moving production of the Focus to China will not result in any job losses for US hourly employees.
Brexit timetable and negotiation plan revealed
Anxious by immigration and loss of sovereignty, Britons voted last June to become the first nation ever to leave the 28-nation EU. Davis said he was looking for a "positive and constructive tone" to deal with the myriad issues dividing both sides.
Supreme Court Will Hear Gerrymandering Case
Four justices - only Justice Clarence Thomas remains of that group - said it was not the court's business to make such decisions. The court will be pressed with the responsibility of deciding if partisan gerrymandering violates the Constitution.
Hike Messenger launches Hike Wallet
The new update has also brought in improved user interface as well as new features to the camera on the app. With Hike Wallet users can also recharge their phones and pay postpaid bills right from inside the app.