Rebels seeking to remove Theresa May risk bringing about “the most appalling chaos”, which could destabilise the country and damage Britain’s international reputation, Jeremy Hunt has said.

Hunt, the foreign secretary and one of the most ambitious of May’s cabinet ministers, chastised his Conservative colleagues for encouraging letters of no confidence in the prime minister.

A predicted coup against May failed to materialise on Monday as it became apparent that hardline Eurosceptics had not yet managed to persuade enough Tory MPs to write to Sir Graham Brady. May would face a confidence vote among the parliamentary party if 48 letters were sent to Brady, who chairs the 1922 Committee of Tory backbenchers.

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A fresh challenge for May came on Monday night as the Democratic Unionist party’s 10 MPs abstained and, in some cases, voted with Labour in a series of votes over the finance bill. Budget measures are covered by the confidence and supply arrangement between the Conservatives and the DUP which the prime minister needs to secure a majority.

The move appeared to be a DUP warning to May that they are not willing to support her administration if she pushes ahead with her Brexit deal, which they have vehemently rejected.

Earlier, Hunt, on a visit to Tehran, said May had “perhaps got the most difficult job of any prime minister or president in the western world at the moment”.

“It is incredibly challenging,” he said. “Somehow she has to find a way of navigating the shoals of Brexit in a way that allows us to have national closure on the issue and move on and deal with all the other issues we face.”

Hunt said May was “by far the best-placed person” to steer the relationship with the EU to ensure the UK leaves on friendly terms. “Seeking to remove her risks the most appalling chaos that could be immensely damaging to our national reputation, but also destabilising and potentially stopping us getting through to the other side of Brexit,” he said. “That is why when people think about this, people will realise that backing her is the right thing to do.”

Downing Street has said significant progress is still needed in the Brexit negotiations to flesh out the final political declaration on the future relationship, including addressing some language on customs that alarmed Brexiters.

Hunt said he had begun talks with European foreign ministers to see if it was possible to “clarify both sides’ intentions” and put minds at rest with the future relationship document. “I do not think you can change any issues of substance because the withdrawal agreement is a legally binding document and the future relationship is a political declaration,” he said. “What you might be able to do is clarify what everyone’s intentions are in the withdrawal agreement where there is inevitably quite a lot of suspicion on both sides.”

May’s spokesman said she would play a negotiating role in the final stages before Sunday’s summit in Brussels, although there is no day set this week for her anticipated meeting with the European commission president, Jean-Claude Juncker.

The prime minister said over the weekend that she intended to go to Brussels this week, before the emergency summit on Sunday. No 10 denied that the threat of a possible no-confidence vote was the reason no date had been finalised.

Responding to the DUP MPs’ decision not to back the government in the budget votes, Jon Trickett MP, the shadow minister for the Cabinet Office, said: “We no longer have a functioning government. With Brexit only a few months away, something has got to give.”

The number of letters publicly submitted to Brady declaring no confidence in May reached 26 on Monday, well short of the tally needed, though Eurosceptic MPs insisted more had been submitted privately.

Sources played down any possibility of a united intervention from the “gang of five” Eurosceptic cabinet ministers, including Michael Gove and Andrea Leadsom, who had hopes of changing aspects of the final agreement.

Critics of the prime minister did little to hide their frustration that a confidence vote had not yet been triggered. Steve Baker, the former Brexit minister and member of the European Research Group (ERG), who has been coordinating the letters campaign, said: “Who knows now? Will MPs do what they said?”

One of the Tory rebels, Simon Clarke, said it was incumbent on those colleagues who had promised to act to do so. “I think they now need to search their consciences and follow up on what they pledged to do,” he said. “If we continue with this plan we are simply not going to have a government. It is quite clear to me that the captain is driving the ship at the rocks.”

Ben Bradley, another MP who sent a letter, said colleagues who had concerns about the deal should know that May had no intention of changing course. “We have two options: carry on as we are, with our heads buried in the sand, reach stalemate, and then general election territory with Corbyn in the ascendency, or do something about it now and change the deal,” he said. “The PM is adamant, she will not change it, so now has to be the time for those colleagues to act.”

At the CBI conference on Monday, May gave a coded warning to her mutinous party, saying Brexit was not an exercise in political theory.

“We have in view a deal that will work for the UK. And let no one be in any doubt: I am determined to deliver it,” she said. “We are not talking about political theory but the reality of people’s lives and livelihoods. Jobs depend on us getting this right.”

May said those rebels threatening to force a no-deal Brexit were costing UK businesses and making them less competitive. “While other countries are forging a competitive future, Westminster seems to be living in its own narrow world, in which extreme positions are being allowed to dominate,” she said.

Downing Street hopes progress on the political declaration could persuade some wavering Brexiters. It is currently seven pages long in draft form, but officials said they expected the final version to be significantly longer. “This is clearly a very intense part of the negotiations,” May’s spokesman said.

One phrase that has particularly alarmed Brexiters in the political declaration says both sides have committed to “combining deep regulatory and customs cooperations, building on the single customs territory provided for in the withdrawal agreement.”

Critics suggested it was a step towards a permanent customs union. A No 10 source said it was one of several areas, as well as security, where there was still work to be done.

The ERG and the Brexiter group Global Britain will convene a meeting on Tuesday morning as May’s cabinet meets, hosted by the ERG chair, Jacob Rees-Mogg, the former Brexit secretary David Davis and Peter Lilley, a cabinet minister under Margaret Thatcher. The groups will publish a paper that they say will “challenge the myths” on the threats to UK trade of leaving the customs union on WTO terms.