This leaves the Prime Minister more or less at the mercy of Brussels, as if they do not accept her Chequers plan she is out of alternative options - except, perhaps, not leaving the European Union at all, which she recently threatened was a possibility if Brexiteers did not back her.
"I would simply say that Brexit can not be and will not be a justification for creating more bureaucracy", he said.
She says this would keep "frictionless" trade and make a border between the Republic of Ireland - an European Union member - and Northern Ireland unnecessary.
He also insisted that the responsibility for resolving any outstanding issues rested with the British, since they were the ones who made a decision 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.
'They're wrong. No bluffing, ' he said. "We need to de-dramatise things", he added.
Meanwhile, the European Commission said on Thursday that the chance of a no-deal Brexit is increasing and that companies and private individuals should prepare for every eventuality.
Conflict between the two regions of Ireland ended two-decades ago, but a bad Brexit deal could make tensions flair.
Conscious of May's fragile political situation in London, EU ministers were keen not to be harsh on the white paper, that they see as part of the next phase of talks on the future partnership, rather than the urgent negotiations needed to finalise the withdrawal agreement.
Media captionTheresa May was speaking during a visit to Northern Ireland on Friday.
"And, on that basis, I look forward to resuming constructive discussions".
"You can't have one side fulfilling its side of the bargain and the other side not, or going slow, or failing to commit on its side".
Mrs May's speech came shortly after one of her aides raised eyebrows by refusing to say the word Fermanagh "in case I mispronounce it".
And she took a swipe at former foreign secretary Boris Johnson's claim - repeated in his resignation speech to the Commons on Wednesday - that technological solutions could be used to avoid the need for infrastructure at the border.
Equally, she said that a hard border between Northern Ireland and the Republic was "almost inconceivable".
Barnier said he had invited British negotiators to work on the backstop issue next week.
"We can improve the backstop", Barnier said.
The Fund's report said a "soft" Brexit scenario - with the United Kingdom out of the customs union but retaining access to the single market and agreeing to abide by EU rules - 'would imply nearly zero cost for the EU as a whole'.
This would be based on the Commission's proposal but "not necessarily" stick precisely to the outline released in March, he said.