Both sides acknowledged the clock was ticking, with the date for the UK's departure from the European Union fixed for March 2019.
But the point was effectively conceded before talks began, with Mr Davis yesterday accepting an European Union timetable set out last week, which makes clear trade will only be discussed once "sufficient progress" is made on citizenship, Northern Ireland and a "single financial settlement" of as much as £88 billion. Today's talks however 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.
Davis said the talks were off to a "promising start" and denied that Britain had caved in on the sequencing of the talks.
European Union officials had earlier warned May not to make that offer so early in the talks.
Adding to the complexity is Britain's current political situation, as May lost her Conservative majority in parliament after a snap election, sparking wide criticism of her leadership and worries about how long May and her cabinet would stay in power.
Merkel also expressed a desire for constructive talks with Britain, but made clear that the EU's priority now was its own future: "We will conduct these talks in a good spirit", she said.
Monday's talks marked the start of what will be one of the most politically sensitive global negotiations of modern times.
After Monday's seven-hour session, European Union chief negotiator Michel Barnier described the meeting as "useful to start off on the right foot as the clock is ticking". And it was useful for me to sit down with my counterpart, David Davis.
"In a first step, we will deal with the most pressing issues".
The Brexit Secretary said: "The position hasn't changed, we have the Lancaster House speech, the two white papers, and the Article 50 letter, all backed up by a manifesto too".
Our aim is to have one week of negotiations every month.
He said "we are witnessing the return of the European Union rather as a solution, not a problem". If its assessment is positive, the talks will then move on to "scoping the future relationship on trade and other matters".
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.
The vote came as a profound shock to Brussels against a backdrop of rising anti-EU sentiment, with many - including now U.S. President Donald Trump - predicting the bloc's eventual break-up.
"They should be agreed alongside each other, this is completely consistent with the Council's guidelines which state nothing is agreed until everything is agreed".
"It's not when it starts it's how it finishes that matters", he said. The EU had no intention of punishing the British despite "a lot of spin" on the United Kingdom side, he added.
"I am not in a frame of mind to make concessions, or ask for concessions. It's not about punishment, it is not about revenge", Barnier said.
May's government said it was "confident it can achieve a bold and ambitious deal that will work in the interest of the whole U.K".