Davis said before the talks began that he is determined to reach a Brexit deal "that works for the whole of the United Kingdom". The insistence of European Union figures and government to agree the "divorce terms" first, means agreeing a full trade deal by March 2019 is now highly unlikely.
Officials on both sides play down expectations for what can be achieved in one day.
Britain agreed on Monday to the EU's timetable for Brexit talks, which states that sufficient progress must be made on divorce issues including citizens rights, the exit bill and Northern Ireland's border with the Republic before negotiations can start on a future trade deal.
Until now, British prime minister Theresa May had insisted that both discussions should be held in parallel.
Frenchman Barnier struck a firm tone as he said the timetable for Britain's divorce after four troubled decades of membership made sense.
In any case, European Union officials say, London no longer seems sure of what trade arrangements it will ask for.
"The protection of the Good Friday agreement and the maintenance of the Common Travel Area are the most urgent issues to discuss", said Barnier, adding that "concrete and imaginative" solutions will be required to control goods and services without creating a hard border.
The two top negotiators from Britain and the European Union struck a conciliatory tone on Monday, after they concluded the first round of Brexit negotiations.
It would offer the United Kingdom control over laws and access to the single market in return for free movement of people.
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 Britain's negotiating position had not changed as a result of his Conservative Party's poor showing in recent elections. "And fair means that we want to keep the British as close as possible to the European Union but never at the price that we divide the remaining 27 European Union states".
"We will then publish a detailed paper outlining our offer on (next) Monday which I believe will form the right basis on which to reach agreement", he said.
"There is a long way to go, but we are off to a promising start".
As Europe grows more confident in its future after the election victory in France of pro-EU President Emmanuel Macron and the growing assurance from German Chancellor Angela Merkel that she will be re-elected in September, analysts say there may be fewer reasons for Europe to compromise. Both sides have set out some principles but this doesn't give us any clarity on how things will look and operate once we leave.
Mr Coveney said the start of Brexit negotiations in Brussels yesterday underlined the urgent need to get devolution back up and running in Belfast.
"We are citizens first".
Barnier has warned that the negotiations must be wrapped up by October 2018 to allow time for all parties to ratify a final accord by March 2019.