"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. The EU has to date rejected a temporary customs union.
The agreement would avoid the need for an Irish backstop - which has left Britain and Brussels deadlocked - and would stop Northern Ireland being treated differently to the rest of the UK.
It came as more than 70 business leaders backed a referendum on the terms of the Brexit deal, warning that the United Kingdom faces "either a blindfold or a destructive hard Brexit" that would be bad for both firms and jobs.
The EU has proposed keeping Northern Ireland inside a customs union with the bloc to remove the need for border checks on the island.
Simon Coveney, the Irish foreign minister, made clear on Monday morning that this would never be agreed.
British Prime Minister Theresa May has secured concessions from Brussels that will allow her to keep all of the U.K.in a customs union with the European Union and avoid a hard border in Northern Ireland, the Sunday Times reported, without naming sources.
In regards to customs checks, the deal is said to include an agreement by the European Union that checks on goods can take place at places of manufacture, or in shops, instead of at the border.
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".
However, the newspaper also reported that the new deal will include an "exit clause" created to convince those who support Brexit that remaining in the customs union is only temporary.
He said there was a "rapidly ticking clock" but that minds were now being focused with a view to concluding a deal and he welcomed that.
"We continue to make good progress in the negotiations but there is work still to do", the prime minister's official spokesman said.
The Sunday Times reports that the prime minister is close to a political deal on the United Kingdom and EU's so-called future economic partnership, which the newspaper says will "allow Britain to keep open the prospect of a free trade deal resembling that enjoyed by Canada".
"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".
He added: "We hope a deal can be done but we're not there yet". "Are we being hurtled toward a Hotel California Brexit where we can check out but never leave?"
"As soon as MPs understand what is really at stake, I have no doubt that they will throw this deal out".