With the two-year clock on Brexit ticking away since March, when a letter from May formally started proceedings, Barnier dismissed suggestion of postponing the negotiations and said such a delay would only prompt further instability.
In the hours immediately after Thursday's embarrassing election setback for the Conservatives, British media reported that finance minister Philip Hammond had told May she needed to put "jobs first" in any new deal with Brussels.
Following London talks with the DUP, May will head to Paris to meet with French President Emmanuel Macron who has just won an impressive victory in parliamentary elections.
It was a surprise move - Mr Gove was sacked as justice minister by Mrs May a year ago after his bid to become party leader forced now-foreign minister Boris Johnson from the race, amid accusations of treachery and political backstabbing.
"The people of Britain have had a bellyful of promises and politicking", he wrote in The Sun tabloid.
"There can be no backsliding from the objectives the PM set out in the [election] campaign - taking back control of our laws, our borders, our cash; but also ensuring that we have a great new partnership with the European Union that allows us to trade more freely and enthusiastically than ever", Mr Johnson wrote in the right-wing tabloid The Sun.
But since coming to power three weeks after the shock vote to leave the European Union, the prime minister has advocated a hard Brexit, which would entail Britain leaving the single market and cutting immigration from the bloc. "This doesn't depend entirely on us", Winterstein told a daily briefing.
Mrs Foster would not go into great detail about the talks but said government stability, Brexit, counter-terrorism and doing what's right for Northern Ireland economically were high on the agenda. "I think most of the Scottish Conservatives will want to see perhaps some changes with the policy going forward".More news: Sessions: Russian Collusion An 'Appalling And Detestable Lie'
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While talks were ongoing, former Conservative Prime Minister John Major told BBC Radio 4's World at One that he was "wary" and "dubious" about the proposed Tory-DUP deal.
On Sunday, she unveiled her full Cabinet, making few changes.
May also restored former Justice Secretary Michael Gove to the Cabinet in another move created to show she was willing to listen to critics.
British Prime Minister Theresa May gives a speech at 10 Downing Street after meeting with the Queen in London, Britain on June 9, 2017.
William Hague, a former leader of the Conservative Party, called for business groups and lawmakers from all parties to be brought in to agree a national position on Brexit.
May has shown little public contrition for the electoral gamble that backfired spectacularly, but was forced to accept the resignations of her two top aides - reportedly a requirement by cabinet colleagues for allowing her to stay in office.
Mrs Foster said: "There's been a lot of commentary around the issues that we are talking about and it won't surprise anyone that we are talking about matters that pertain, of course, to the nation generally".
Defence Secretary Michael Fallon has also sought to downplay the DUP's future role, saying the government was not looking at a formal coalition but would seek assurances that the DUP would vote with May "on the big things".