In a withering assessment of Mrs May's plan on Friday, Barnier ripped up her proposals on customs arrangements, including collecting tariffs and resolving the issue of the Irish border.
He also insisted that the responsibility for resolving any outstanding issues rested with the British, since they were the ones who chose to leave - and that they would have to be resolved probably by December, as the process of ratifying any withdrawal agreement in the European Council, European Parliament, and British Parliament would not allow for negotiations to carry on to March 2019, when Brexit is supposed to happen - on paper, at least.
'We're going to have to do a reset and come back and look at it all again, ' he said.
The issue had yet to be formally discussed among those involved in the EU-UK negotiations, but informal discussions suggest there is limited appetite for such a development, sources in Brussels added.
But while the negotiations have rarely looked so in danger of failing, senior EU officials and diplomats do not see any objective in an extension to article 50, the two-year negotiating period set aside for a member withdrawing from the union, unless there is a change in British politics that can offer genuine hope of a better outcome.
In a speech to politicians and business leaders in Belfast's docklands, May accepted a hard border between Northern Ireland and the Irish Republic once Britain leaves the bloc would be "almost inconceivable", but dismissed the EU's current plan as "unworkable".
Mrs May has consistently said that she would never accept such a proposal and believes no British Prime Minister could ever accept it.
He warned there would be "conditionality" under the Article 50 withdrawal mechanism between settling Britain's exit payment and creating a new relationship with the EU.
"We need to find the right place and right time to carry out checks to protect consumers and to protect the single market", he said.
Mr Barnier told reporters yesterday that it was "a matter of urgency to agree a legally operative backstop", saying: "We need an all-weather insurance policy".
Barnier accepted that the White Paper had brought about "constructive discussions" and they are "not that far" from reaching an agreement.
The Guardian understands that during a meeting of European Union ministers with Barnier on Friday, astonishment was expressed at May's comments in Belfast earlier in the day in which she accused the European Union of seeking to constitutionally and economically "dislocate" the UK.
Foster's political party has 10 voting members in the lower house of British parliament, and is working towards a pro-British and pro-Irish Brexit deal.
The customs partnership plan was published by the United Kingdom in a long-awaited Brexit white paper last week.
The minister said the economic impacts of Brexit are "very considerable" and it is important that engagements are intensified between the British government and the European Union taskforce.
"Right across the United Kingdom, far more unites than divides us - perhaps the greatest strength of our union is its potential", she added.