Meanwhile, Mr Barnier said he believed a "fair" deal was possible for both sides but he made clear the talks, beginning a year after the UK's Brexit referendum, would take place according to a timetable set by the EU.
"We must first tackle the uncertainties caused by Brexit", Barnier said, citing the rights of European Union citizens in Britain and the possible impact on the open border between Northern Ireland and Ireland.
Anxious by immigration and loss of sovereignty, Britons voted a year ago to end their country's four-decade membership in the 28-country bloc - the first nation ever to do so.
Mr Davis will be accompanied by a nine-strong negotiating team that includes the most senior civil servants at the department as well as officials from the Treasury and Home Office as well as Mark Sedwill, the national security adviser to the Prime Minister.
"Given the range of political and technical issues at stake, I think this very much justifies our dialogue and the dialogue should indeed represent a safeguard to all concerned that we are very much committed to this as a priority", Mr Barnier said.
A look at what is ahead now that Brexit talks have started
So, we each have to assume our responsibility and the consequences of our decisions. "The consequences are substantial".
Both men said they had spent a great deal of time discussing the question of Northern Ireland where all parties seek to uphold the free border.
Merkel spoke after meeting with Romanian President Klaus Iohannis, who also stressed the 27 European Union countries' unity.
May had promised to take Britain completely out of the bloc's common trading area and slash the number of people coming from the EU.
The talks will be in English and French, they said in a joint statement. As in any divorce, count on both sides to be picky in splitting the goods and dues.
While the European Union negotiating team led by Michel Barnier has been ready for months, Britain stalled even after it triggered the two-year process on March 29.
But Brussels has ruled out any consideration of those issues until progress has been made on its key demands -Britain's exit bill, estimated by Brussels at around 100 billion euros ($112 billion); the rights of three million European Union nationals living in Britain and the one million Britons on the continent; and the status of the border between Northern Ireland and Ireland.
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It comes as Theresa May's cabinet is torn over how to approach negotiations with Chancellor Philip Hammond and others pushing for a softer approach, while Brexiteers threaten to resign if the Prime Minister abandons her tough stance.
Britain now appears to have given in on the EU's insistence that the negotiations first focus on three key divorce issues, before moving on to the future EU-UK relationship and a possible trade deal.
"The best way we can spend this week is to rebuild trust", rather than tackle the big hard issues right at the start, a European source said.
"This is a technically hard issue, but it's one which I am certain is soluble, although it will probably take us to the end of the process, when we have already decided what our customs and free trade arrangements are", Mr Davis said.
"We're leaving the EU and because we are leaving the EU, we will be leaving the single market and by the way, we will be leaving the customs union".
"The new Northern Irish executive needs to be set up in a few days' time, at the same time we have a new government and a new Taoiseach in Dublin and of course there are the ongoing political discussion in London which we are also following closely".
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