But Mr Barnier said: "Our backstop can't be extended to the whole UK. Why?"
Underlying the debate is the goal of ensuring that the border between Northern Ireland and the Republic of Ireland remains open, with no border posts or customs checks, after Brexit to protect economic links and peace on the island.
The arrangement gives terminals and freight operators a measure of short-term certainty, according to BPA chief executive Richard Ballantyne.
In a coded warning to pro-EU Labour MPs, Starmer said there were "very divided views" in the party about a Norway-style model and he was "injecting some honesty about where we are in the party".
May hopes to resolve the border issue with a wider trade deal between the European Union and Britain, but has agreed to the need for a plan B if this is delayed or does not happen. "How does that fit with the absence of a hard border in all circumstances?" "We are no closer to knowing the relationship Britain is going to have (with the EU)", said Jane Foley, a currencies strategist at Rabobank, adding that many investors were waiting it out in the meantime.
The DUP, meanwhile, will not accept the EU's preferred option of creating a customs border in the Irish Sea, as it threatens the "constitutional integrity" of the union.
Mrs May said the backstop would only be necessary if "for technical reasons" the new customs arrangements were not up and running by January 1, 2021 when the transition period expires.
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".
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The Democratic Unionist Party accused Mr Barnier of lacking respect for the constitutional integrity of the United Kingdom, accusing him of "an outrageous attempt to revert to the annexation of Northern Ireland". Critics say that is vague, and European Union leaders have repeatedly said Britain can't have the benefits of membership without the responsibilities.
Mr Barnier said he continues to have "great respect" for the prime minister.
It followed speculation he was ready to quit the government over the details of the backstop Northern Ireland border proposal.
Friends of Mr Johnson said it was disappointing that the private dinner had been covertly recorded but senior Conservative Sarah Wollaston said dressing up the comments "under the cover of a "private" discussions won't wash".
An anti-Brexit group backed by billionaire financier George Soros said on Thursday it would target lawmakers in an attempt to strengthen its campaign for a new referendum on Britain's departure from the European Union.
The development came as Mrs May braced herself for possible defeat in the Commons next week on an amendment which would require her to try to negotiate a permanent customs union with the remaining EU.