One option being considered by the British government is a "customs partnership" that would see the United Kingdom collect tariffs on the EU's behalf at ports and airports.
This would involve the United Kingdom acting on the EU's behalf when imports arrive from the rest of the world.
Downing Street has spent the last few days intensely lobbying ministers including David Davis, Liam Fox and Boris Johnson on the benefits of the so-called hybrid model, under which the United Kingdom would collect European Union import tariffs on behalf of Brussels.
"In order "to give ourselves the flexibility to negotiate a deep [US] trade agreement, ' we were told that the United Kingdom would need to remove itself 'from the regulatory sphere of the EU" and align with the US's regulatory model".
According to The Guardian, however, Mrs May will attempt to stave off rebellion by delaying the final decision on which customs model to go for.
"We would have external duties set by the European Union", he said. But the assumption is that it would, which would mean a lot of additional work in big ports such as Rotterdam. No-one really knows, but it would take a long time to set up any new system of this kind. The "Smart Border 2.0" report, which is often cited as an option, has rightly been acknowledged as insufficient by the Government.
Writing in the Belfast Telegraph, Mr Barnier said he was looking for "practical solutions to avoid a hard border" and his priorities in such a deal were protecting the Good Friday Agreement, ensuring north-south cooperation (including on economic matters), and respecting the UK's constitutional order.
The report says that a wide-ranging free-trade agreement with the United States could boost Britain's economy by 0.35% of GDP after Brexit.
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Brussels is apparently concerned both by the prospect of "porosity" on the EU's external border and by the risk of creating a precedent which might be copied elsewhere, while Tory Brexiteers fear the scheme could indefinitely trap the United Kingdom within the EU's customs arrangements, as well as being expensive and complicated to operate.
Tory Brexiteers fear the scheme could indefinitely trap the United Kingdom within the EU's customs arrangements, as well as being expensive and complicated to operate. There are fears that any infrastructure introduced at the border between Northern Ireland and the Republic of Ireland could destabilize the Peace Process.
The Lords' amendments to the European Union (Withdrawal) Bill can be overturned in the lower house, where May has a working majority with the support of a small Northern Irish party, albeit a slim one.
The letter says it must be straightforward in future for schools to recruit from European countries, and that those already working here must have their rights protected in a post-Brexit visa regime.
Meanwhile, Brexiteer plans for a looser customs arrangement were said to be gaining support despite warnings they were "playing with fire".
It's also worth remembering that there is one other large fly in the ointment here - the whole border debate isn't just about customs and tariffs.
"It won't be ratified, because they can't sign a deal with us until we are a "third country", so that will be shortly after the formal departure from the Union". So, even the most innovative customs system in the world doesn't get rid of the need for border checks entirely.