Davis insisted that talks about trade would occur "in parallel", but Barnier said they would start only in a second phase.
Davis said both sides were "off to a promising start" even though the challenges ahead were daunting.
Barnier said there was agreement that the negotiators would first look at citizen's rights, the outstanding bill Britain must pay for previous European Union commitments and the Irish border issue.
Vulnerable to mutineers in her own party and to a newly empowered opposition, while struggling to negotiate a parliamentary partnership with a hardline party from Northern Ireland, May introduced the speech by stressing she would respond "with humility and resolve" to the electorate's message.
Only when "sufficient, concrete progress" on the first phase has been made will Mr Barnier recommend to the European Council that the negotiations can enter the next stage, taking in the future trading relationship, with that recommendation possibly coming at October's summit of EU leaders.
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".
Both sides acknowledged the clock was ticking, with the date for the UK's departure from the European Union fixed for March 2019.
While Barnier insists on the "sequencing" of talks, so that trade negotiations can not start until probably January, finding a way to avoid a "hard" customs border for troubled Northern Ireland may well involve some earlier discussion of the matter.
Merkel said Monday: "I think it is premature to speculate on the first day of the negotiations how they will end".
Modi thanks those who participated in Yoga Day worldover
Many countries which do not know our language, tradition, or culture, are now connecting to India through Yoga. The event saw around 3,000 participants who took part in the session and performed Yoga.
"The UK has been crystal clear in our approach to the negotiations, the withdrawal process can not be concluded without the future relationship also being taken into account".
"The position we have agreed today is completely consistent with our long-term position".
In remarks that have come back to haunt her, she also warned voters: "Britain simply will not get the right Brexit deal if we have the drift and division of a hung parliament".
"So it's the same as it was before".
"I am not going to say we are going to make concessions", Mr Barnier said Tuesday.
"It's not about punishment, it is not about revenge". He said that the other European Union countries have a united position but the British are "in chaos". "It is not the other way around.So we each have to assume our responsibility and the consequences of our decision, and the consequences are substantial".
UK Prime Minister Theresa May will outline on Thursday her approach to the "hugely important issue" of reassuring European Union expatriates about their futures in Britain at a dinner for European Union leaders in Brussels on Thursday night.
Asked if he had made any concessions to the United Kingdom in exchange, Barnier said the talks were not about "punishment" or "revenge" and the two sides had to simply accept the "responsibility and the consequences" that came with their decisions.
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