May refused to give a "cast-iron guarantee" that the cut-off date for the backstop plan would not extend beyond December 2021, but said that the government expects to have a better customs arrangement in place by then "at the very latest".
Brussels had demanded that Northern Ireland remain aligned with European Union customs regulations once Britain leaves, in a bid to prevent a hard border appearing between Northern Ireland and the Republic.
He backed down from threats of resignation after the PM offered an olive branch over the backstop solution, insisting the Government expects the deal to come to an end by December 2021.
Barnier hit out at Britain's pro-Brexit politicians who accuse the European Union of taking a tough stand that slows the pace of negotiations.
Liberal Democrat leader Sir Vince Cable said the apparent Cabinet disarray over the customs backstop made a damaging "no deal" Brexit more likely.
But, with separate talks also starting on how a future EU-UK trading relationship may work, negotiations are deadlocked over how to achieve that while neither leaving a back door for British goods to enter the EU nor raising new hurdles for commerce between Northern Ireland and the British mainland.
The UK's proposed "backstop" plan for trade with the European Union after Brexit was published after an "expected" end date - of 2021 - was included in it.
A source close to Davis told HuffPost UK: "Obviously there's been a back and forth on this paper, as there always is whenever the government publishes anything".
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.
He added, "We will not leave this issue unresolved and need it resolved by the autumn". He wanted it to be time-limited.
But he said: "Theresa May and her team have agreed to the backstop in the March agreement and there is no question of backtracking on that".
In a pointed reference to the absence of Brexit secretary David Davis and his negotiators from Brussels for much of the first half of 2018, he said his team were "happy and ready to step up the frequency of our discussions".
He also said that it was hard to see how a customs arrangement could be extended to the full United Kingdom. Does it respect the integrity of the single market/customs union?
The Cabinet Office's Northern Ireland border document states: "The UK is clear that the temporary customs arrangement, should it be needed, should be time limited, and that it will be only in place until the future customs arrangement can be introduced".
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