Mrs May is relying on the support of Arlene Foster’s DUP to prop up her unstable Government in what is known as a “confidence and supply” deal.
But the unionists are warning of a revolt if Mrs May agrees to the terms the European Union are calling for.
The bloc’s chief negotiator Michel Barnier has said there could be a deal by next Wednesday but only on the condition the Prime Minister agrees to install a border between Northern Ireland and the UK.
He said: “The UK wants to leave, and will leave, the single market and the customs union.
“This means that there must be checks on goods travelling between the EU and the UK – and these checks do not exist today.
“We have agreed with the UK that these checks cannot be performed at the border between Northern Ireland and Ireland.
“Companies in the rest of the UK would fill in their customs declarations online and in advance when shipping goods to Northern Ireland.
“The only visible systematic checks between Northern Ireland and the rest of the UK would involve scanning the barcodes on the lorries and containers, which could be done on ferries or in transit ports.”
But the DUP has stipulated a firm red line in regards to a border being instated, as the party endeavours to retain close ties with the United Kingdom after it exits the EU next March.
A DUP source told Sky News: “It is unacceptable that we would be treated differently to the rest of the UK.
“We will not be bounced into anything.
“If Theresa May doesn’t take our concerns on board, she may not be the leader to take us through Brexit.”
Treasury Minister Mel Stride responded: “The Prime Minister’s position has always been crystal clear on this issue of no border down the Irish Sea – we have said it many, many times.”
Ms Foster, who has previously referred to the union between England, Wales, Scotland and Northern Ireland as her party’s “guiding star”, has said the unionists are prepared to vote down the budget in protest in the event a Brexit deal is struck which crosses her party’s red line.
Budget motions are generally seen as a vote of confidence, which means the ominous threat from the DUP could cast doubt on Mrs May’s future.
The DUP has continually declared it would veto any Brexit deal that created a border down the Irish Sea, which is precisely what Mr Barnier is now saying would be necessary to carry out checks on goods passing between Northern Ireland and the rest of the UK.
Mr Barnier told businesses in Brussels he wanted goods to be registered online before transportation.
He also called for customs inspectors on ferries crossing the Irish Sea.
The proposals form part of the so-called “Irish backstop” arrangement, which is intended to come into effect if the UK and EU do not agree on alternative arrangements to avoid a hard border.
Ms Foster warned yesterday she couldn’t accept a deal which featured “customs or regulatory barriers within the UK internal market”.
She added: “The ties between our two economies are so intricately linked that to put up barriers, tariff or none, would be economically disastrous for Northern Ireland.”