The close alignment envisaged under the deal would remove the need for a "backstop" for the Northern Ireland border. Is the future of the union stake?
Reports of a secret Brexit deal come just as more than 70 business leaders signed a letter calling for a second referendum after warning that a bad deal could threaten the United Kingdom jobs industry.
Glen, the financial services minister, said he is "extremely confident we will reach an imminent deal", Reuters reported.
The Sunday Times reported that the EU would allow the creation of a whole-UK customs union that would avoid the need for a Northern Ireland border "backstop" that has been at the heart of the impasse in negotiations.
Speaking ahead of the publication of Mr Johnson's comments, a Downing Street spokeswoman said: "The Prime Minister is clear we are leaving the customs union".
Mr Lidington replied by saying: "We're certainly. very close to resolving it".
"Obviously still having this issue in relation to the insurance arrangements for Northern Ireland and Ireland, and that very much remains our focus and attention in getting that deal".
"There is no competitor worthy of replacing us", he added.
"We continue to make good progress in the negotiations but there is work still to do", the prime minister's official spokesman said.
The Taoiseach also said the UK's withdrawal from the bloc had undermined the historic Good Friday Agreement that ended three decades of violence in Northern Ireland and called for a guarantee that there would be no return of a hard border with the Republic.
Ireland and the European Union will "never" accept a time-limited Brexit backstop, the United Kingdom has been told.
He said: "It doesn't solve a fundamental problem, which is that if the agreement is still based on the Chequers plan - if it involves the so-called common rule book i.e. regulatory alignment of all sorts of manufactured goods between the European Union and United Kingdom - it will not satisfy the Brexiteers in any way and will be a crucial stumbling block for her".
"This has been committed to by the United Kingdom in order to have a Withdrawal Agreement".
"I think it's reasonable for us to expect a country like the UK and a government like the UK Government to stand by its commitment".
Chief Executives from Waterstones, Innocent Drinks and Lastminute.com stated that the United Kingdom faces either a "blindfold or a destructive hard Brexit" in the note.
More than 70 business leaders - from lastminute.com founder Martha Lane Fox and former J. Sainsbury PLC chief Justin King to Cobra Beer founder Karan Bilimoria and ex-chairman of Rolls-Royce Holdings PLC Simon Robertson - signed a letter arguing that both the government's current plans for Brexit, and a no-deal Brexit, would be bad for companies and jobs.