"If you ask me are we going to make concessions, I must tell you that it is the United Kingdom that is leaving the EU, the single market, the customs union and not the other way around", the former European commissioner and French foreign minister told a joint press conference with Davis.
Without a majority for her Conservatives in the U.K. Parliament, May now says she will work to reflect the wide range of views on Europe that exist in Britain, a signal her Brexit policies could soften.
Barnier said after welcoming Davis that it was important to tackle uncertainties caused by Brexit.
The bloc would first seek to reach an agreement on European Union citizens rights, the UK's departure bill, and the future border between the Republic of Ireland, an European Union member, and Northern Ireland, a British region.
Roth said that "Brexit is a very, very hard operation" and there's only a bit over a year to negotiate it.
But the business community and many lawmakers want to retain closer ties with Europe, and they are heaping pressure on the prime minister to change her approach.More news: Bodies found on 'ripped' USA destroyer
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.
Davis and Barnier will hold a press conference later on Monday. "And fair means that we want to keep the British as close as possible to the European Union but never at the price that we divide the remaining 27 European Union states".
The Brexit Secretary read out a section from May's Article 50 letter, which only seemed to highlight his concession, before adding: "When we eventually get to the point where the Council decide we have made enough progress, both sets of dialogue will continue, including free trade, not just trade, including customs, security, obviously cooperation in all sorts of ways".
European stocks rose on Monday, partly on optimism about the talks actually getting underway after months of sniping and uncertainty, analysts said. At a meeting of European Union foreign ministers in Luxembourg he said: "The most important thing for us is to look to the horizon, raise our eyes to the horizon".
Prior to the General Election the previous May government outlined a desired outcome involving leaving the single market and customs union, ECJ control and EU payments, as well as restricting EU immigration (though welcoming the "brightest and best") while at the same time aiming to protect existing expats and obtain a favourable deal for trade. More than 80 percent of them voted in the election for a party that advocated ditching the single market, while 77 percent told polling company YouGov that they wanted out. Austrian Foreign Minister Sebastian Kurz said: "If we don't succeed, both sides will lose".
The EU says it will not compromise on its core "four freedoms": free movement of goods, capital, services and workers.
It comes as Philip Hammond warned failing to secure a Brexit deal would be "very, very bad" for the country and insisted there must be transitional arrangements to avoid a "cliff edge".