Mrs May will brief fellow European Union leaders at a summit on Thursday on the UK's approach, which will be set out in a paper on Monday, Mr Davis revealed.
The Brexit talks were finally launched at the European Commission's headquarters in Brussels almost one year after Britain's June 23, 2016 referendum to leave the bloc.
Representatives for the United Kingdom (U.K.) and the European Union (EU) kicked off Brexit negotiations in Brussels on June 19, nearly 12 weeks after Britain's prime minister triggered the mechanism that would let it depart from the EU.
Despite Theresa May wanting parallel talks on a new trade deal while negotiating the "divorce", this would have to wait until later, the European Union said.
"The EU 27 position is clear in terms of what conditions we'd like to see for our citizens there and what we can offer for United Kingdom citizens here", the diplomat said. "I look forward to working closely with you during this negotiation", EU negotiator Barnier said, as quoted by the press service of the European Commission, the EU's executive.
An early election this month, in which British Prime Minister Theresa May lost her Conservative majority in parliament, only added to the problems.
In choreographed talks that saw the two men exchange mountaineering gifts, they agreed to discuss divorce issues before negotiations on a future trade deal can start.
"In a first step, we will deal with the most pressing issues".
The document released after the first day of discussion on "Terms of Reference for the Article 50 TEU negotiations" stated: "The following initial negotiating groups have been established: Citizens' rights; Financial Settlement; Other Separation issues". David Davis and I, as chief European Union negotiator, will discuss the issues together, tackle difficulties, and lift obstacles.
Queen to outline UK government agenda in scaled-down speech
Arlene Foster's party backs Brexit, but wants to avoid any disruption to movements across the border with the Irish Republic. May's authority is badly damaged as Britain begins negotiations on leaving the European Union.
Mr Barnier said: "There is a very, very sensitive political dimension to this".
Though less visibly upbeat than veteran Brexit campaigner Davis, Barnier insisted the two sides would work together for a "fair deal" that would not "punish" Britain.
"We've laid solid foundations for future discussions and an ambitious but achievable timetable", said David Davis, Britain's Brexit minister, adding he was "encouraged" by the first talks.
"I am not in a frame of mind to make concessions, or ask for concessions", he said.
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.
"Third, it may show that the United Kingdom (unlike in May) is less able to negotiate the UK's future trade relationship with the European Union, as the the United Kingdom government since the election does not have an agreed position".
Ireland has been part of the European Union since 1973 and has had a free-flowing border with Northern Ireland since the Good Friday Agreement in 1998 that ended 30 years of sectarian violence in Northern Ireland, known as "The Troubles".
"It's not about punishment, it is not about revenge".
"I would like us to get a good agreement that is in both sides' interests".
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