Barnier said there will be one week of negotiations every month and the two sides will use the time in between to work out proposals.
The EU and Britain on Monday launched the long-awaited Brexit talks at European Commission's headquarters in Brussels, almost one year after Britain voted to leave the bloc by a narrow margin on June 23, 2016.
London agreed that "English and French will be used as working language".
"It was clear from the opening that both of us want to achieve the best possible outcome and the strongest possible partnership".
The talks began nearly a year to the day after Britain shocked Europe by voting to cut loose from the 28-nation European Union, its biggest market.
"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". He had struck a conciliatory tone in an earlier statement, emphasising Britain and the EU's "shared European values".More news: West Indies annouces 13-member squad for 1st two ODIs against India
After seven hours of talks in Brussels, Mr Davis - who had previously promised the "row of the summer" over the timetable for the negotiations - said he was optimistic about the talks.
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.
It will test the ingenuity of thousands of public servants racing against the clock to untangle 44 years of European Union membership before Britain is out, 649 days from now, on 30 March 2019.
The EU says it will not compromise on its core "four freedoms": free movement of goods, capital, services and workers. But the Brexit bill may be fudged, with a final figure not emerging until 2018, to help the government sell it as part of a new trade deal.
He and others have also raised the possibility that Britain could remain in 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.More news: Special Counsel Is Reportedly Investigating Jared Kushner's Business Dealings
May is now seeking backing from the DUP, a Northern Irish party, for her minority Conservative government after she lost her parliamentary majority in a June 8 election.
The center-left Social Democrat strongly criticized May's Conservatives, saying that they "played with the emotions of citizens in Britain, told fake news about Europe and left people unclear about what consequences this would all have". "A fair deal is possible and far better than no deal", the French former European Union commissioner said. "That is why we will work all the time with the United Kingdom, and never against the United Kingdom".
European Union officials believe, however, that May and her ministers are coming round to accepting Brussels' rules. "But we are definitely ready for it because Europe is weaker without the British but I think the British would also be weaker without us Europeans".
"I think it was recognition by all sides that the clock is ticking and we do really need to push on with this now and start to make positive strides towards getting a deal that is in both sides' interests", the source said. The election of the fervently europhile Macron, and his party's sweep of the French parliament on Sunday, has revived optimism in Brussels.
But he said that Monday morning's terror attack in London and the devastating fires in Portugal reminded him that "there is more that unites us than divides us".More news: Google leverages search engine expertise to launch robust employment engine
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