Mr Thompson said last month the so-called "max fac" plan to use modern technology to solve the Irish border question would leave firms facing huge charges for customs declarations and for European Union "rules of origin".
The move came after Brexit Secretary David Davis was reported to be considering resigning unless there was a clear time limit on the temporary customs arrangements which would be adopted if there is no final Brexit deal.
The EU proposed keeping Northern Ireland in the customs union but this was rejected by the government as well as Northern Irish party the DUP, which is holding together Mrs May's minority government.
The future of the border between European Union member Ireland and Northern Ireland after Brexit is one of the most hotly contested issues in the negotiations, with many on both sides of the border wanting to keep the current free-flowing frontier open.
Barnier said the two sides had made progress but that a lot remains to be done, particularly on three subjects which, he said, were "very serious for business and citizens".
Britain may hope that this wording might still win approval from officials in Brussels, and May's spokeswoman said it was a "matter for the negotiations" as to whether the date became part of a legal text for Britain's withdrawal.
The Brexit Secretary is set to return to Brussels next week for talks with the EU's chief negotiator Michel Barnier.
While 42.1% of respondents said that they would vote to join the Republic of Ireland if there was a referendum border, a further 45% said that they would rather remain in the United Kingdom. We will not accept such a proposal.
"It would tie us effectively into the EU's customs arrangement for an indefinite period", he said.
Redmond O'Neal Charged with Attempted Murder After Vicious Crime Spree
According to police, "the suspect became angry and repeatedly punched the victim in his head and threw him on the ground". The spree came to an end when O'Neal was arrested for armed robbery at a convenience store in Santa Monica on May 8.
Mr Barnier also rejected the idea of a time limit on the backstop which Downing Street has requested, an issue that almost led to the resignation of Brexit Secretary David Davis.
The pound slipped to the day's low of $1.3371 in the hours after the proposal's publication.
"There's plenty of Brexit news, but irrespective of the news flow we are still in the same position".
Lord Malloch-Brown, a former United Nations deputy secretary general, said the "chronic uncertainty" surrounding Brexit talks was harming business and it was "time to settle this once and for all".
Less than half of all people in Northern Ireland consider themselves British, according to a new survey.
A day after British Prime Minister Theresa May said she could keep the whole United Kingdom within the European Union customs area for up to an extra year after Britain leaves in March, Barnier dismissed any time-limit.
But John Longworth, of the Leave Means Leave campaign, said Mrs May appeared "obsessed with the damage-limitation mantra of the Remainers" and was leading the country into a Brussels trap.
Time is pressing in the Brexit talks, which have all but stalled as May tries to overcome the divisions not only among her ministers but also in her Conservative Party.