However, this would effectively shift the customs border to the Irish Sea, something which the prime minister said she could not accept.
On the other hand, she has hardcore pro-Brexit Tory MPs, according to The Times, warning that a vote of no confidence could be on the horizon if she reneges on her promise to leave the customs union.
The ministers believe that the so-called customs partnership put forward by May is unworkable and is encouraging Brussels to press for Britain to stay in a customs union after Brexit. But being in a customs union for which Britain has no formal role in setting the rules and tariff schedules is nothing like being in the EU.
But such parliamentary arithmetic means it may be hard to avoid this customs union fight being seen as a de facto vote of confidence. And threatening to remove the whip from rebel MPs could be a self-defeating step for a government without a majority.
After a week in which EU negotiators were said to have given a "systematic and forensic annihilation" of Theresa May's proposals for a new customs relationship to avoid a hard border in Northern Ireland, the United Kingdom government has sought to reassure pro-Brexit Tory MPs that the United Kingdom "will not be staying in the customs union or joining a customs union", according to a Downing Street source.
But speaking on a local election campaign visit, Mrs May insisted that she would not change course.
Those plans included two options, either a "customs partnership" effectively collecting duties for Brussels for goods arriving in the United Kingdom but intended for European Union markets or a "highly streamlined" arrangement making use of technology and regulatory co-operation.
"That report was an absolute mystery to me and I note that it quickly disappeared from the bulletins", the spokesman said.
The question of whether to leave the customs union goes to the heart of the kind of Brexit the United Kingdom will get.
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'Coming out of the customs union means that we will be free to have those deals, deals that suit the United Kingdom, ' she told the BBC on a local election campaign visit to a firm in the West Midlands.
"But I also recognise the importance to businesses like this of being able to have as frictionless a border as possible into the European Union".
Ministers have argued that the move is unnecessary because United Kingdom citizens will automatically receive those protections when Britain leaves the EU.
The Government has suffered three more heavy defeats in the Lords over flagship Brexit legislation.
Several senior Tory rebels helped inflict the defeat - by a majority of 71 - including former Cabinet ministers David Willetts, Michael Heseltine and Chris Patten.
It follows defeats for Mrs May during the passage of the EU (Withdrawal) Bill over membership of the customs union and protection of workers' rights after Brexit.
The amendment was led by independent crossbencher Lord Pannick, supported by an alliance of Labour former Attorney-General Lord Goldsmith, Lib Dem frontbencher Lady Ludford and the Tory John Gummer.
He said: "This Bill should not be used as an excuse to reduce the legal rights which we all enjoy against the state".
On Monday, Justice Minister Lord Keen of Elie sparked heckles as he hit back at the amendment's backers during the debate, declaring that the adoption of a "body of foreign law" would be "one of the greatest constitutional outrages since 1689".