Irish Taoiseach Leo Varadkar told the Prime Minister on Monday that he was ready to consider a review mechanism as part of a "backstop" arrangement to keep the border with Northern Ireland open after Brexit.
The aim would be to satisfy Ireland, which insists that there must be no border infrastructure with the North after Brexit; the Northern Irish DUP party, which props up May's government and insists it must not be treated differently from mainland Britain; and the "Brexiteers" in May's party who say Britain must have the right to do its own trade deals after Brexit.
"The (British) Prime Minister raised the possibility of a review mechanism for the backstop", the spokeswoman added.
The report claims an "all-UK customs deal" will be written into the legally binding withdrawal agreement, which would do away with the need for the controversial "backstop" arrangement agreed by the United Kingdom last December, which would see Northern Ireland remain in full alignment with the EU's single market and customs union rules in the event of a no-deal Brexit.
Cabinet members are due to discuss the status of the Brexit talks on Tuesday morning at their regular weekly meeting, but No 10 was earlier indicating that any update for senior members of the British government was not likely to be substantive because insufficient progress had been made.
The Sunday Times reported that preparations for a final deal were far more advanced than previously disclosed and could pave the way for a special UK/EU summit later this month and a possible vote in Parliament in December. The EU has to date rejected a temporary customs union.
The Irish border has proved the biggest obstacle to a deal, with both sides vowing not to create a hard border between Ireland and Northern Ireland for fear of destabilising the peace accord that ended decades of deadly sectarian violence.
The Times also said Mrs May is on course to get an agreement on a future economic partnership that will let Britain keep open the prospect of a similar free trade accord to the one Canada has with the European Union - a move it said could help persuade the euro-sceptic Tories to back the deal.
The call came after Mr Varadkar dismissed the idea of a time-limited backstop, saying it would not be worth the paper it is written on.
"We have made good progress but clearly in relation to the backstop there are outstanding and significant issues".
Now, top European Union bureaucrats are indicating they are prepared to offer the Prime Minister an "independent mechanism" by which Britain could trigger a "review" on the customs arrangement and move towards ending them.
"I certainly hope we are".
But the government's Brexit department stated that they are confident there will be a deal that works for businesses - and reiterated their stance against a People's Vote.