The Commission said on Friday: "The opening of negotiations at political level next week will focus on issues related to citizens' rights, the financial settlement, the Northern Irish border and other separation issues, as part of the sequenced approach to the talks. Everything has to be behind us otherwise there will be a more brutal exit".
"They think Britain is being somewhat in denial of reality at the moment", said Pierre Vimont, a former high-ranking French diplomat who is now a senior fellow at Carnegie Europe, a Brussels-based think tank. There is a steady dialogue between the two sides that has never stopped at any point.
A Conservative source said there was so far no deal to announce and that a decision on the timing of any announcement would have to wait until an agreement was finalised.
"We never put timescales on when we expect a deal to be done and I'm not going to start now". Her Cabinet - as is her parliamentary party - is sharply split between hard and soft Brexiters.
Speaking as he arrived for a meeting of European Union finance ministers in Luxembourg, he said: "As we go into that negotiation, my clear view - and I believe the view of the majority of people in Britain - is that we should prioritise protecting jobs, protecting economic growth and protecting prosperity as we enter those negotiations and take them forward".
"But we have to be honest, it will take much more than that for us to be convinced that the DUP tail is not wagging the Tory dog", he told reporters. However, the prospect of a deal has prompted warnings that it could upset Northern Ireland's fragile peace.
Sinn Fein's Stormont leader Michelle O'Neill said: "I will be making it very clear that any deal between the Tories and the DUP can not be allowed to undermine the Good Friday and subsequent agreements". "You folks here make enough mess of your own elections, make enough mess of your own governments, make enough mess of your own affairs".
For his turn, Macron wished "negotiations on the exit from the European Union and then on the future relations with the United Kingdom will start as soon as possible".
Talks with the DUP broke up on Tuesday night without an agreement, but Mrs May said the discussions had been "productive".
May's programme will most probably have to be watered down, dropping some of her preferred reforms to help get legislation through parliament and possibly having to give way to other ministers who have strong views over the direction of Brexit. "It's probably as good as we could have imagined".
As May attempts to cobble together a majority, the EU's Barnier said he will hold talks with British envoy Olly Robbins on Tuesday to organize the negotiations.
May has dismissed calls to resign following the dismal election result after calling a vote three years early in the hope of bolstering her slim majority ahead of the Brexit talks.
Brussels has previously warned the UK Government it will only enter talks on a potential future trade deal with Britain once "significant progress" is made on divorce matters. "As the European Union has itself said, 'nothing is agreed, until everything is agreed, '" they added. Even if politics has been extraordinarily volatile over the past year, a reversal of the Brexit vote - which would nearly certainly need another referendum - has an nearly negligible chance of occurring.
The party - which was the Conservatives' junior coalition partner from 2010 to 2015 - created one of the surprises in last week's snap election, increasing their number of MPs from nine to 12.