Number 10 has insisted the Government wants to make sure there is no hard border in Ireland.
It follows the leak to Sky News of a letter from Boris Johnson to the Prime Minister in which he says "it is wrong to see the task as maintaining 'no border'" on the island of Ireland after Brexit - and that the Government's task will be to "stop this border becoming significantly harder".
Its revelation has prompted calls for Theresa May to consider the position of her Foreign Secretary, while Labour have been granted an urgent question in Parliament about future border arrangements in Ireland.
It could see Mr Johnson summoned to the House of Commons to explain the document to MPs.
Writing to the Prime Minister in his letter, Mr Johnson also sought to play down the "exaggerated impression" of "how important checks are" at EU external borders.
He also went as far as contemplating a hard border between Northern Ireland and the Republic of Ireland, writing: "Even if a hard border is reintroduced, we would expect to see 95% + of goods pass the border [without] checks."
It suggests a significant downgrading of expectations about the change to the Irish border as a result of the Government's Brexit strategy and Mrs May's plan to leave the EU's single market and customs union.
As recently as November last year, the Foreign Secretary told the House of Commons: "There can be no return to a hard border. There can be no hard border.
"That would be unthinkable, and it would be economic and political madness".
In February 2016, ahead of the EU referendum, Mr Johnson promised the Irish border would be "absolutely unchanged".
The document is the Foreign Secretary's response to the Prime Minister asking him to substantiate a claim made at a meeting of her Brexit "war Cabinet" on 7 February, when ministers' talks centred on Northern Ireland and immigration.
Referring to the meeting in his letter, Mr Johnson wrote: "I offered, and you [the Prime Minister] agreed, to send you a paper setting how I believed the Ireland/ NI border issues could be managed on the basis of a highly facilitated solution."
The letter from the Foreign Secretary, entitled "The Northern Ireland/Ireland border - the Facilitated Solution", accompanies a "concept note" that "draws on Foreign Office expertise".
This appears to be an attempt to meet commitments given by Mrs May to the EU as part of the first phase of Brexit negotiations, before an agreement was reached in December last year.
Then, Mrs May undertook that, post-Brexit, there would be no hard border between Northern and Ireland and the Republic of Ireland.
In the note, officials appear to have distanced themselves from the conclusions of the Foreign Secretary, stating at the top of the paper: "It is an FCO concept note based on discussion with other departments but is not agreed technical advice."
The thrust of the Foreign Secretary's point to the Prime Minister is that there already is a border on the island of Ireland, so it is "of course a fallacy" that Brexit will see a border re-emerge.
Mr Johnson says HMRC "only" checks 4% of consignments arriving in the EU at the UK's external border, and other authorities only 1%.
He claims "in short the EU border is already largely a border" that relies on electronic paperwork and "not physical inspection".
But the figure quoted is believed not to take into account the much higher level of checks required on agricultural and food trade across that border, in a situation where regulations and standards have diverged.
Labour's shadow Northern Ireland secretary Owen Smith branded Mr Johnson's leaked comments to the Prime Minister as "reckless and irresponsible".
He said: "The Prime Minister should condemn these remarks immediately and seriously consider the position of her Foreign Secretary."
Shadow foreign secretary Emily Thornberry accused Mr Johnson of lying to the House of Commons and to the UK public.
"Is there no issue that he doesn't take seriously? Is there no issue that he thinks that he can't just get through by lying?," she told Sky News.
She added: "The British Government have promised the people of northern and southern Ireland that there will be a soft border and now privately he's saying that actually it probably will be a hard border but 'hey, never mind'.
"This is not acceptable, this is not grown-up politics, this is irresponsible politics and it is simply telling lies to the British people."
Tory peer and former deputy prime minister Lord Heseltine said Mr Johnson's leaked comments are "completely incompatible" with what he said in the EU referendum campaign.
He told Sky News: "What he's now saying to the Prime Minister in private is there may well have to be checks of perhaps 5% and that means a hard border.
"This is the most remarkable revelation of duplicity."
The leak comes on the eve of the publication of a draft legal text from the phase one Brexit agreement, consented to by EU27 officials, with hardline language expected from the European Commission.
It is likely to focus on the "backstop option" of "full alignment" of Northern Ireland with the relevant rules of the EU's customs union and single market, in the absence of solutions such as those offered by Mr Johnson.
EU sources have told Sky News the detail of the legal text, to be published on Wednesday, is required because of a lack of trust in Brussels that the UK is taking the Irish border issue and its commitments seriously.
On Tuesday, the Foreign Secretary was heavily criticised for remarks comparing the Irish border to the levying of the London congestion charge as vehicles travel between the capital's boroughs.
Sky News has contacted the Foreign Office for a response.