The key sticking point remains how to avoid customs checks taking place at the border between Northern Ireland and the Irish Republic, without putting up new barriers between Northern Ireland and the British mainland.
"Brexit has undermined the Good Friday Agreement" - the 1998 peace deal that ended three decades of violence in Northern Ireland - "and it is fraying relationships between Britain and Ireland", Varadkar said.
However, if these Brexiteers are not convinced that the customs union is only a temporary measure they would nearly certainly rebel.
Irish deputy prime minister Simon Coveney said that a time-limited backstop will never be agreed to by Ireland or the EU.
May's spokesman refused to confirm the report, calling it "speculation", according to Sky News.
A Downing Street spokesman said the prime minister and the Taoiseach agreed that the intention is that the backstop should only be temporary and that the best solution to the Northern Ireland border would be found by agreeing a future relationship between the United Kingdom and the EU.
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".
According to the Sunday Times, the private concessions will prevent the need for Northern Ireland to be treated differently to the rest of the United Kingdom - and avoid an Irish backstop.
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EU law expert Professor Steve Peers told this newspaper: 'I think there became an increased chance of a deal once the EU apparently agreed to a UK-wide customs union with the EU in the withdrawal agreement'.
The bloc has also accepted that regulatory checks on goods can take place "in the market" by British officials, rather than at ports by European Union inspectors, according to the Times.
A spokesperson said: "The prime minister has been clear that we are making good progress on the future relationship and 95pc of the Withdrawal Agreement is now settled and negotiations are ongoing".
The paper also reported that the Prime Minister was on course to agree a future economic partnership that would leave open the possibility of Canada-style free trade deal sought by Brexiteers.
Preparations for a final deal were far more advanced than previously disclosed, the report said.
"In March the United Kingdom agreed this backstop will apply "unless and until" a close future relationship eliminates any need for border infrastructure or related checks and controls".
The UK would effectively become a non-voting member of the European Union, forced to accept laws made in Brussels with no power to shape or amend them, he said.
Indeed, some of those officials believe a November summit is essential if there is to be sufficient time to get any deal passed through parliament. "Are we being hurtled toward a Hotel California Brexit where we can check out but never leave?"