Mr Davis said: "It was clear from the opening that both of us want to achieve the best possible outcome and the strongest possible partnership one that works for the United Kingdom and for the EU".
May had called for an early election in the hope of getting a stronger hand to drive a harder bargain in the Brexit negotiations.
A spokesman for the Department for Exiting the European Union said: "We believe that the withdrawal process can not be concluded without the future relationship also being taken into account".
The report did warn against the government targeting an arbitrary limit of absolute numbers of migration, such as a pledge to bring immigration into the tens of thousands.
"That position hasn't changed, it's the same as it was before", said Davis, a tough-talking figure in the "Leave" campaign in last year's Brexit referendum.
Both Davis and Barnier appeared to acknowledge they have a mountain to climb in the talks - by giving each other hiking-related gifts.
"In the first step, we will deal with the most pressing issues".
Mr Davis's agreement to Monday's agenda led some European Union officials to believe that Ms May's government may at last coming around to Brussels' view of how negotiations should be run.
Britain's political instability has caused concerns in European capitals hoping to get the negotiations over with quickly and cleanly, as has May's oft-repeated threat to walk away from the talks without a deal if necessary.
The Chancellor says no deal would be "very, very, bad" for Britain.
"My clear view, and I believe the view of the majority of people in Britain, is that we should prioritise protecting jobs, protecting economic growth, protecting prosperity as we enter those negotiations and take them forward", he said.
They were joined by senior aides as they tucked into Belgian asparagus with Vinaigrette, red mullet with vegetables and fondant potatoes, vacherin (meringue cake) with wild strawberries, mocha coffee and cakes.
Those issues are Britain's exit bill, estimated by Brussels at around 100 billion euros (us$112 billion), the rights of three million European Union nationals living in Britain and one million Britons on the continent, and the status of the border between Northern Ireland and Ireland.
"I think it's premature to speculate about the outcome on the first day of talks", she told reporters in Berlin.
Davis said the talks would be carried out in "a positive and constructive tone", with Britain looking to forge a "strong and special partnership for the future".
Residents now enjoy free movement across the border and many businesses have facilities on both sides.
"The best way we can spend this week is to rebuild trust", rather than tackle the big hard issues right at the start, another European source said. An attempt to strengthen her hand by calling an election backfired and she's run into further domestic strife since, while the 27 other European Union members started out with more leverage anyway.
Before the election, May proposed a clean break from the European Union: leaving its single market, which enshrines free movement of people, goods, services and capital, and proposing limits on immigration and a bespoke customs deal with the EU.
The aftermath of the inconclusive vote has left Britain's prime minister scrambling to try and form a power-sharing agreement with the Democratic Unionist Party in order to form a minority government.