Downing Street has said Theresa May continues to have full confidence in Boris Johnson, after the Foreign Secretary attacked proposals for a customs partnership after Brexit as "crazy".
Mrs May is understood to favour the arrangement, under which the UK would collect customs tariffs on behalf of the EU, as a means of breaking the deadlock in Brexit talks on the future of the Irish border.
But she failed to win over senior colleagues at a meeting of her Brexit "war cabinet" last week, forcing her to ask officials to rethink the plan, along with a second "maximum facilitation" option using new technology to reduce friction at the border.
In what was being seen as a very public challenge to the Prime Minister's stance, Mr Johnson used an interview with the Daily Mail to warn that the customs partnership option would create a "whole new web of bureaucracy".
The plan would not comply with promises to take back control, and would hamper the UK's ability to strike trade deals, said the Foreign Secretary.
Honoured to be the first foreign minister to visit @SecPompeo in Washington, and the first to sign his visitors' book! Evidence of the enduring strength of the UK-US relationship. pic.twitter.com/iDAf5XY7TW
-- Boris Johnson (@BorisJohnson) May 7, 2018
"It's totally untried and would make it very, very difficult to do free trade deals," he told the Mail.
"If you have the new customs partnership, you have a crazy system whereby you end up collecting the tariffs on behalf of the EU at the UK frontier."
Mrs May's official spokesman said that the issue was not discussed at Tuesday's regular meeting of Cabinet in 10 Downing Street, which Mr Johnson attended after returning from a visit to the US.
The spokesman declined to say whether the PM had spoken privately with the Foreign Secretary about his comments.
But asked whether Mrs May continued to have full confidence in Mr Johnson as Foreign Secretary, the PM's spokesman said: "Yes."
He added: "There are two customs models that were put forward by the Government last August and most recently outlined in the Prime Minister's Mansion House speech which the entire Cabinet was signed up to.
"Following last week's sub-committee meeting, it was agreed that there are unresolved issues in relation to both models and that further work is needed.
"The Prime Minister asked officials to take forward that work as a priority."
The chair of the European Research Group of Tory Eurosceptics, Jacob Rees-Mogg, who has previously described the customs partnership as "cretinous", said that Mr Johnson had "hit the nail on the head".
But Liberal Democrat leader Sir Vince Cable said the semi-public spat made clear the depth of internal divisions on Brexit within the Cabinet and made it less likely the Government would be taken seriously by Brussels chief negotiator Michel Barnier.
"Yet again this Tory Government is demonstrating to the Europeans who are negotiating with them that they are internally divided," said Sir Vince.
"When a leading Tory describes the Prime Minister's proposals as "cretinous", why would Mr Barnier take them seriously?"
Labour MP Rupa Huq, a supporter of the People's Vote campaign for a second EU referendum, said: "Boris Johnson's description of Theresa May's preferred option of a 'customs partnership' as a 'crazy system' is typically intemperate but rooted in truth.
"The Prime Minister's plan would replace today's frictionless trade with a bureaucratic nightmare of tracking goods not just at the border but more or less anywhere and everywhere they are stored or distributed."
But Ms Huq said that Mr Johnson's preferred "Max Fac" scheme would also raise costs for businesses and consumers, adding: "To put jobs first, to protect the Irish peace process and to give our young people a future we should stay in the customs union and single market and put any final Brexit deal to a People's Vote."
Conservative former attorney general Dominic Grieve said Mr Johnson's decision to speak out was "regrettable", but said he could understand why Mrs May was willing to put up with his "outbursts".
Mr Grieve told BBC Radio 4's World at One: "I don't think he is in any way inhibited by normal propriety in government.
"I can well understand that seeing the difficult issues that we are having to confront, which are very divisive, the Prime Minister should accept these rather extraordinary bursts of misbehaviour by Boris."
Business Secretary Greg Clark stressed on Sunday that thousands of British jobs depend on frictionless trade with Europe, in what was viewed as an attempt to revive the customs partnership model.
The row came as Mrs May faces two more parliamentary defeats on her Brexit plan as it nears its final stages in the House of Lords.
Peers want to remove the Prime Minister's planned exit day of March 29 2019 from flagship legislation that takes Britain out of the European Union.
A second amendment would allow EU laws to be replicated in the UK and allow future participation in its agencies.
Both proposed changes to the EU Withdrawal Bill have cross-party support in the upper chamber, which means they are likely to win in a vote.
Meanwhile, a further change has been proposed to the legislation that would require the Government to negotiate continued membership of the EEA.
Labour peers will be ordered to abstain because it goes against official party policy.