The European Union is proving unable to convince Theresa May that by using “trusted trader schemes” and technology its proposal to in effect keep Northern Ireland in the customs union and single market will not draw a border in the Irish sea.

The Brexit negotiations have reached an impasse over the failure to find an acceptable solution to avoiding a hard border on the island of Ireland after the UK leaves the EU.

The solution proposed by Brussels in which Northern Ireland has a different status from the rest of the UK has been rejected by the prime minister as involving the economic and constitutional “dislocation” of the country.

The EU’s chief negotiator, Michel Barnier, has nevertheless repeatedly insisted that the issue can be “de-dramatised”.

Barnier has sought to show that the level of trade between Belfast and the rest of the UK is minimal, and that the checks that would be required do not pose a constitutional threat to the British government.

But according to what is described as a diplomatic note seen by the Times, the EU is struggling to convince the UK that it is significant that checks at a border could be avoided entirely for many companies through trusted trade schemes.

The diplomatic note, said to have been drafted following a meeting of EU ambassadors last Wednesday with Barnier’s deputy, Sabine Weyand, reports that the UK has not been “helpful” over the issue.

The note says: “The biggest unsolved problem is Northern Ireland. There is a political mobilisation in the UK in this regard. Therefore, we are trying to clarify the EU position.

“The controls or checks only have to be organised in a way that would not endanger the EU single market. For the main part, these controls would not have to happen at a border. It is to be expected that the reach of the backstop would decrease anyway in case of an agreement about the future relationship … The solution is specifically phrased for Northern Ireland so that it is not applicable for Scotland. A UK concern.”

“In these talks the UK is not at all helpful,” the note said. “They are now working on a new version of the protocol themselves in which for the most part the words ‘Northern Ireland’ would not be mentioned.”

The notes adds: “The EU wants a transit procedure or process. Controls could happen at the companies and they would just scan a label on board ships.”

As first reported by the Guardian, Barnier has become increasingly frustrated with the failure of Dominic Raab, the Brexit secretary, at one point complaining that the UK was withholding data requested over the summer by the EU’s Brexit taskforce.