The UK agreed to settle the issue of citizens" rights, borders and the so-called "divorce bill' with the European Union before discussions over trade begin, when formal Brexit negotiations finally got under way in Brussels on Monday.
Frenchman Barnier struck a firm tone as he said the timetable for Britain's divorce after four troubled decades of membership made sense.
Mr Barnier spoke at a press conference after the first day to confirm that settling priority issues would build "trust on which to negotiate a new relationship".
"We acknowledge that, as Britain leaves the EU, there are genuine and reasonable concerns among our EU colleagues about oversight of financial markets that will then be outside EU jurisdiction". But it must be done in a way that works for Britain ...
"There may well be the need for certain transitional periods, some of them which will involve phasing out, others potentially phasing in", Barnier told reporters in Luxembourg after a meeting of European Union affairs ministers. A fair deal for both sides was possible, he said.
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Britain, the EU's biggest financial market, was leaving the bloc in 2019, raising the prospect of an abrupt cut in cross-border links without a new trade deal.
Davis said Britain's negotiating position had not changed as a result of his Conservative Party's poor showing in recent elections. With now only 21 months to sign and ratify the exit deal, there is a limited time for any trade agreement.
The election had been called by the Conservative Party with the intention of consolidating its hold over United Kingdom politics.
In his first major comments in six weeks, he also said he wants to see how the economy responds to the "reality of Brexit negotiations".
In Berlin, German Chancellor Angela Merkel emphasised the unity of the remaining 27 European Union countries, who have been alarmed in recent weeks by May's threats to walk out of the talks. "But the 27 of us will formulate our interests very clearly and hopefully, together", Merkel said.
"So we must be clear about ours".
Macron, a committed pro-EU leader and ally of Merkel, also easily won French legislative elections on Sunday, cementing his power base.
It means the divorce terms will be set first, before discussions on future trade relations.