As the Brexit pressure ratchets up another notch, Michel Barnier appears everything most of the current British government is not: clear, calm, precise – and logical to a fault.

“I promised myself from the start,” the EU’s silver-haired, grey-suited chief negotiator told a packed theatre near the Gare du Nord in Paris on Saturday, “that I would not allow passion or emotion into my approach to Brexit.

“I work on facts, on figures. On what is legal and operable. My obligation is to defend, calmly and firmly, the interests of the European Union, of its citizens, its companies, its regions … And in leaving, the UK cannot ask us to change what we are.”

If that was the view from the EU side, then from Downing Street it was precisely the opposite.

At a meeting on Friday evening the prime minister’s closest adviser, Dominic Cummings, was demanding that the EU stop being so stubborn and show flexibility. His message was that if Brussels did not budge in Brexit talks, Boris Johnson would not fold but would simply take the UK out without a deal.

“Next week we are going to know how things turn out,” Cummings said. “If the EU says no then we are not going to do what the last lot did [he meant Theresa May] and change our negotiating position. If we don’t get anything next week we are gone.”

With just weeks to go before the UK is due to leave the EU on 31 October, there seems to be total deadlock – and the blame game has already begun.

Barnier was clear on Saturday that a no-deal outcome would be Johnson’s responsibility, while the prime minister and Cummings have their strategy planned well in advance – to lay responsibility for that outcome and any turbulence it could cause firmly at the door of the EU.

Last week Johnson’s government outlined its proposals for a Brexit deal which rests on Northern Ireland leaving the EU customs union on 31 October, but remaining in the single market for goods, and the Northern Ireland assembly being given the right to veto the arrangements every four years.

Barnier says the proposals are not acceptable in anything like their current form. “If they do not change I do not believe, on the basis of the mandate I have been given by the EU27, that we can advance.”

The differences over customs and borders went to the very heart of what the EU was. “We are a single market. That’s a complete ecosystem, with common rights, common norms, common standards, common rules, a common legal system. It requires checks at its borders.”

In Westminster, though, Johnson has worked hard to rally his own divided party behind his plan – and for now it is holding firm.

Writing in the Observer, the culture secretary, Nicky Morgan, who voted Remain and has previously strongly opposed a no-deal outcome, urges the EU to show willing. “They need to look seriously at this proposal, talk to us, and negotiate a deal that works. This is the best chance we have to break the deadlock and leave with a fair and reasonable deal.”

Both sides are entrenched and few see how a breakthrough can now be achieved before the crucial EU summit on 17-18 October.

If Johnson were to give ground to the EU now, he would risk losing the alliance of support he has assembled in parliament that includes the DUP and anti-EU Conservative MPs.

Were he to give in over customs arrangements or the “Stormont lock”, which gives Northern Ireland’s assembly the right to decide every four years if it wants to continue to apply EU legislation to traded goods, he would face the same opposition that defeated Theresa May’s deal.

Nigel Evans, executive secretary of the 1922 committee and a hardline Brexiter, said: “The kind of concessions that the EU wants over a customs union and the single market would leave us with Brexit in name only.

“I don’t think for one minute that the prime minister would be willing to make such compromises but if he did he would start to lose the DUP and a very large part of the ERG.”

Johnson now seems willing and ready to drive a no deal through. That outcome will, however, present him with another huge problem. If there is no deal by the end of the EU summit, he will find himself bound, under the terms of the Benn act, to ask the EU for an extension to the UK’s membership until 31 January next year.

Boris Johnson may be ready to push a no-deal exit through, but will be constrained by the Benn act to ask for an extension. Photograph: Henry Nicholls/Reuters

Downing Street says he will not break the law, but it also says Brexit will not be delayed. Johnson’s insistence that he will not ask for an extension was contradicted by a submission to the court of session in Edinburgh that was made public on Friday, in a case brought by campaigners seeking to force Johnson to comply with the Benn act. The submission appeared to show that the government accepted it would have to make the request.

One possible way for the prime minister to square the circle – to write the letter yet avoid a delay in the event of no deal – would be to persuade one or more EU leaders to veto an extension, once a letter had been sent. The Hungarian prime minister, Viktor Orbán, a rightwing eurosceptic, is the name most often mentioned. But few in the EU think even Orbán would risk helping out Johnson in this way. A senior Tory minister remarked: “There are EU budget negotiations coming up and Hungary needs funding. Orbán would be putting all that at risk.”

The consensus among senior Tory and Labour MPs this weekend is that for all Johnson’s bravado, an extension is the most likely outcome. That might look in the short term like failure for Johnson, but there are plenty of Tories who believe it would work to his advantage in a subsequent general election. “He will say he was the prime minister who took on the EU, parliament, and the courts to deliver the will of the people but they all stopped him. That strategy of the heroic struggle could well work in the end.”

For Barnier, it is all clearly hugely frustrating. After three years’ work, he says, “we found those solutions. Last November, with Theresa May. We worked, seriously, methodically, together with her government.” And now there is Boris Johnson “questioning a very important part of that agreement”. Johnson, he says, must realise that “a deal is between two parties”. Britain cannot demand concessions the EU cannot make, as the argument goes round in circles.