Barnier told the EU’s 27 ambassadors on Sunday that a weekend of discussions had not yet produced a breakthrough, according to a diplomatic note of a briefing that took place on Sunday evening.

He said that talks with the UK had focussed on customs arrangements, with Britain now asking to distinguish between goods from Great Britain that enter Northern Ireland and remain there, and ones that go onto the EU.

Under Johnson’s latest proposals, Northern Ireland would legally remain in the UK’s customs union, but would at the same time continue to abide by the EU’s rules on customs and tariffs, avoiding border checks in Ireland. The arrangement would in effect create a dual customs system that would see goods tracked and businesses having to apply for a rebate to cover any differences in tariffs.

One of several reservations raised by Barnier is the difficulty of knowing whether a good ultimately ends up in the EU’s market and supply chains.

Despite an apparent success in striking the basis of an agreement with Varadkar, the UK side is concerned that the European Commission will effectively bring talks to a close this week by insisting that the full Northern Ireland-only backstop — which would see Northern Ireland effectively remain in the EU’s customs territory — must be added as an underlying insurance policy if the UK is to leave the EU on October 31.

EU officials are highly sceptical that an agreement on a new text can be found before leaders meet in Brussels at the end of next week for a European Council summit. The UK accepting a Northern Ireland-only backstop is seen as a more feasible solution.

Government officials say that if the EU insists on the full Northern Ireland-only backstop, it will be unacceptable to the UK and has no chance of securing a majority in the House of Commons.

The backstop is the legally operative insurance mechanism included in the withdrawal agreement that guarantees that the border between Northern Ireland and the Republic of Ireland remains open after Brexit in all circumstances, while also protecting the integrity of the EU’s single market. Johnson has pledged to ditch it, while the EU insists that any alternative needs to achieve the same effects and be legally operative from the day Brexit happens.

One option under consideration in Number 10 is that, if the EU accepted in principle Johnson’s customs partnership for the Northern Ireland proposal without a backstop, but said it needed more time to iron out the details, Johnson could agree to a three-month technical extension to Article 50.

Some senior Tories believe that Johnson could present this outcome to the public as the prime minister successfully “getting Brexit done” by October 31, with a deal agreed in principle and the backstop removed. A technical extension would be used to negotiate the final details and the UK would formally leave at a later date.