Brussels wants a backstop that would mean Northern Ireland remaining in the customs union as well as large parts of the single market.
There is no hard border at present, as goods and services are traded between the two countries with few restrictions as both are now part of the EU single market and customs union.
"We are working not just with the domestic preparations that we need to do, but with the Belgians, the Dutch and the French, to make sure not just passengers going, as you say, on holiday but also goods can continue to flow".
The EU has consistently blocked British attempts to agree on a finish date for the "backstop" that will keep the whole United Kingdom in a customs union, and now many Conservative Brexiteers are arguing a potentially unlimited time frame is unacceptable, the BBC reports.
The Europeans want it to apply "unless and until" alternative arrangements are agreed after Brexit, with the understanding that if a future UK-EU economic partnership either proves elusive or requires systematic checks on trade, the backstop could remain in force indefinitely.
Mrs May's counter-proposal is for a "temporary customs arrangement" for the whole United Kingdom, but Tory Brexiteers are suspicious this could turn into a permanent situation, restricting the freedom to strike trade deals around the world.
The need to resolve the backstop issue is a political headache for the Prime Minister, who depends on the votes of the 10 DUP MPs to prop up her administration in Westminster.
British Isles brace for Storm Callum as powerful gusts hit Ireland
The alerts mean flooding is possible rather than expected, with high river levels and standing water on roads likely. SE Britain will miss the worst of the weather although the rainband will finally clear eastwards during Sunday.
They added that to wait until Tuesday's Cabinet meeting and somehow "bounce" ministers into supporting her deal would be "the worst possible way of doing it".
But the proposal has many opponents before it gets approved.
Theresa May attempted to head off Cabinet resignations from pro-Brexit ministers by insisting she would not allow Britain to become "trapped" in a customs arrangement with the European Union.
He added: "We are simply reminding her, you are being torn at the moment between what we believe are your gut instincts and your sound principles on the Union and at the same time the bullying, threatening behaviour of the EU".
But British and European Union negotiators are making headway on the Irish border issue, the biggest hurdle to an overall agreement, and hope for a Brexit deal breakthrough on Monday, diplomats said.
Meanwhile the difficulties the prime minister will have in pushing any deal she may secure through parliament are highlighted by new analysis that suggests she will have to rely on the support of more than a dozen Labour MPs to avoid her Brexit deal being torpedoed.
He told BBC's Newsnight Brexit was a "fundamental, existential issue" and "on that basis maybe some people will obviously consider their position".