The EU says the United Kingdom needs to come up a workable alternative to their proposals - which it describes as a "backstop".
In publishing on Wednesday its first draft of a withdrawal treaty which both sides hope to agree by October, the European Union said it would be happy to consider that.
She reiterated her commitment to avoiding a hard border between Northern Ireland and the Republic, but also emphasised a border in the Irish sea breaking up the UK's common market would be unacceptable.
All sides have pledged to avoid a hard border including customs checks, in order to protect the 1998 Good Friday agreement in Northern Ireland, which ended three decades of bloody sectarian violence.
The Good Friday deal saw checkpoints dismantled and the border is now open, ensuring the free movement of goods and people, with some 30,000 crossing it every day.
The EU therefore set out in detail the third option whereby Northern Ireland would retain the EU's rules on goods, sanitation and agriculture, environment rules and the customs code.
"I think we are working against the clock so I think it would be hard for those technologies to be developed by the end of the transition period", said Magdalena Frennhoff Larsen, lecturer at the University of Westminster. FoodDrinkEurope wants to "maintain the status quo for as long as necessary, allowing existing trade and customs arrangements to continue largely unchanged until a new [comprehensive] trade agreement enters into force", a spokesperson said.
May is due to set out her plans in a long-awaited speech on Friday, which her spokesman said she "hoped that European leaders would engage with this thinking constructively".
Mrs May has said her long-term goal is a "bespoke economic partnership", underpinned by a comprehensive free trade agreement guaranteeing tariff-free access to European Union markets for British goods and services.
At odds with current left-wing Labour leader Jeremy Corbyn and his popularity hobbled by anger at his role in the 2003 Iraq invasion, Blair said "a comprehensive plan on immigration control, which preserves Europe's values" would be "right for Europe and timely for the evolving British debate on Brexit".
"This is very much a default and would only apply should it prove necessary".
The Tories rely on the DUP for votes, and the government has responded in a strongly worded letter to Tory MPs, following the publication of the draft withdrawal agreement Wednesday.
"It is our British responsibility to find one - we created the problem, not the European Union", he said in London.
But Kim from Keynsham, who has family in Northern Ireland, believes Theresa May purposefully didn't address the issue in her speech.
European Union leaders have been pushing British Prime Minister Theresa May to provide more details about how she sees the border between Ireland and Northern Ireland operating, how long a transition period should last to help ease Britain out from next year and to explain her government's vision of their future relations.
Paul Breen, a lecturer at the University of Westminster, said May was involved "a political 'Game of Thrones'".
"Every time the Cabinet meets, all we get is even more freaky soundbites. And it will be a disaster", he said.