Irish Prime Minister Leo Varadkar said he was open to creative language and creative solutions to the issue, but reiterated Ireland's view that the border backstop clause in a Brexit deal can have no expiry date or unilateral exit clause.
But a spokesman for Mrs May said no extra meeting has yet been scheduled ahead of the regular cabinet meeting in Downing Street next Tuesday.
Britain and the European Union both wish to keep the border between EU-member Ireland and Northern Ireland in the United Kingdom open after Brexit, as it is seen as crucial to the 1998 Good Friday peace accord that ended decades of sectarian bloodshed in Northern Ireland.
Sources close to talks said London regarded it as a "big step" that the European Union side now appears ready to contemplate a means for bringing the backstop to an end short of a broader trade treaty.
Mr Varadkar's spokesman said the Irish premier indicated he was ready to consider proposals for a review mechanism, but only if it was clear that the arrangement can not be ditched by one side acting alone.
The prime minister's official spokesman cautioned that even though there were rumours of a breakthrough earlier in the week the deadlock drags on.
"While the United Kingdom should aim to conclude the Withdrawal Agreement as soon as possible, this would not be done at any cost".
"The prime minister said that, once agreement was reached on a withdrawal agreement, it remains the case that nothing is agreed until everything is agreed and it will be subject to securing an acceptable full future framework".More news: Democrats projected to get 238 seats in House of Representatives
The Sunday Times report said preparations for a final Brexit deal were "far more advanced than previously disclosed" and that May's agreement would satisfy both remain-voting Tories and the hardline Eurosceptics within her party.
However, if these Brexiteers are not convinced that the customs union is only a temporary measure they would nearly certainly rebel.
The Downing Street spokesman said the cabinet "needs more time" to consider mechanisms to ensure Britain can not be bound to the EU.
DUP MP Sir Jeffrey Donaldson said Dublin's stance was making a no-deal Brexit likely.
Michel Barnier has warned that a Brexit deal is "not close" and challenge Thereda May to make the tough choices necessary to get it over the line in the coming weeks. Instead, May's team is aiming to have a deal ready for the end of the month, but that will require the political will to get over the hurdle of the Irish border question, the person said.
'So, we think there's a deal to be had if they recognise that the deal is unacceptable to Parliament, I think that opens up a vista of the opportunity of the real negotiations'.
However, once they return to Northern Ireland pet owners will have to go through a similar process as the certificates do not allow pet owners to move freely between north and south of the border.
All pet owners are now being told to ensure they have the correct health protection documents in place for the possibility of a no-deal Brexit.