"There is a unity of objective among people in the United Kingdom", May said, following a meeting with French President Emmanuel Macron in Paris.
Mr Barnier held "talks about talks" in Brussels with Mrs May's Brexit adviser Olly Robbins and Britain's ambassador to the EU, Tim Barrow, on Monday but they failed to agree on a date for the negotiations to begin.
Last evening, May faced the tough 1922 Committee of her party's backbench MPs amid a brewing rebellion within the ranks after her gamble to call a snap general election backfired, leaving the Tories eight seats short of a majority.
Her failure to win a majority has put May under pressure over her Brexit plans from inside and outside her party and has prompted complaints about her choice of partner due to the DUP's stance on social issues such as gay marriage.
Mrs May's rivals have also warned the government's claimed impartiality as a mediator in the Northern Ireland political dispute would be fatally undermined by the DUP pact.
Before the government can do anything it must finalise a deal with the Northern Irish Democratic Unionist Party (DUP). These include a commitment by London to do more to attract Foreign Direct Investment into the province, improvement in connectivity between Northern Ireland and Great Britain - including a guarantee of direct access to London's hub airports.
British Prime Minister Theresa May has called on the European Union to issue a sweeping, reciprocal guarantee of rights for British expatriates living in the European Union after Brexit.
Justin Thomas sets US Open record with 9-under 63
With a blind shot to the green, he went long into deeper hay, and his first hack at it moved the ball only a few feet. A whopping eight players are at 6 under, including world No. 4 Hideki Matsuyama and Louis Oosthuizen.
Sinn Féin's seven newly-elected MPs were in London on Tuesday for a round of meetings, although they will not be taking up their seats at Westminster. "This new arrangement is very unsettling and people are concerned and anxious about what it may mean, or what promises may be given".
However, some positives could come out of the potential deal, Barry McElduff, the newly elected Sinn Fein MP for West Tyrone said.
"We stand at a critical time with those Brexit negotiations starting only next week - I think that stability is important. I can't negotiate with myself", he told The Financial Times.
A day after France and Germany both said Britain could reverse Brexit if it wanted, Verhofstadt said Britain's budget rebates and opt outs from European Union rules would disappear if it did so.
Guy Verhofstadt, the European Parliament's chief Brexit negotiator, warned Tuesday that "the current uncertainty can not continue" and on Wednesday issued five "pressing questions" on Twitter.
Manfred Weber, the German head of the centre-right European People's Party, the largest group in the assembly, said May now has "no majority" for a so-called hard Brexit, in which Britain leaves the EU with no deal on the divorce terms or on future relations with the EU.
The chancellor, Philip Hammond, is preparing to fight for the United Kingdom to remain within the EU's customs union which could safeguard jobs and trade with EU members but would severely restrict the UK's ability to strike its own trade deals around the world.
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