It would limit Britain's ability to do post-Brexit trade deals, Johnson warned.
Environment Secretary Michael Gove risked fuelling the row after describing as "helpful" a Twitter thread by former aide Henry Newman - now director of the Open Europe think tank - which described resurrecting the customs partnership as "surely misguided".
Customs union members can not negotiate their own trade deals outside the EU, which is why leaving it - while hopefully negotiating a bespoke arrangement - has been one of the government's Brexit goals.
All EU members are part of the customs union, within which there are no internal tariffs (taxes) on goods transported between them.
He insisted that the EU would defend Irish interests as fiercely as Dublin, but the Democratic Unionist party leader, Arlene Foster, said the EU was being unrealistic, while Britain's former ambassador to Washington, Sir Christopher Meyer, accused the EU of putting the Good Friday agreement at stake by "weaponising" the border issue.
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But critics argue this will place Britain too close to the current custom union arrangements and make it hard to conduct its own trade deals post-Brexit.
A number of Cabinet ministers spoke out against the "customs partnership" idea which would see Britain collect tariffs on behalf of the European Union for goods destined for the block, with firms potentially claiming back a rebate if products remained in the United Kingdom on a lower-tariff regime.
Most importantly, he said the plan - which will leave the United Kingdom collecting tariffs on behalf of Brussels while operating its own independent regime - did not amount to "taking back control".
Johnson, an outspoken supporter of Britain's withdrawal from the European Union, said the plan backed by Prime Minister Theresa May would not fulfil numerous promises of Brexit.
Mr Rees-Mogg, chairman of the European Research Group of Tory MPs, dismissed Mr Clark's warnings about the impact of rejecting the customs partnership.
A spokesman for the Prime Minister pointed out that the Foreign Secretary had twice given his backing to the "customs partnership" option, with the most recent occasion being just two months ago ahead of her Mansion House speech. The so-called "maximum facilitation" arrangement "would make an unequivocal break with the European Union regime", says The Guardian.More news: 1 killed, 2 badly wounded in South Africa mosque attack
Theresa May subsequently asked her officials to draw up "revised proposals".
Could the issue topple May?
In a scathing assessment of the Foreign Secretary's behaviour, Mr Grieve concluded: "I don't think he is in any way prohibited by normal propriety in government".
The Business Secretary, highlighting the importance of having a customs deal with "the minimum of frictions" to firms operating a just-in-time production line, noted how Toyota, which employs 3,500 people at plants in the United Kingdom, was deciding where its next plant should be in Europe.
"Following last week's sub-committee meeting, ..."
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Eurosceptics urged Ms May to drop the proposal for a "customs partnership", which they fear would tie the United Kingdom too closely to Brussels.
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