"It is important that Britain would not undercut our own products on our own market in the all-UK Irish backstop", one source said of the European Union demand that comes as the two sides are seeking a Brexit deal as soon as this month.
But the DUP has interpreted the wording of her letter to mean that the measure will be contained in the Brexit divorce deal despite Mrs May's insistence it will never come into effect.
Such a move would avoid a hard border on the island of Ireland.
DUP leader Arlene Foster said the letter "raised alarm bells for those who value the integrity of our precious Union and for those who want a proper Brexit for the whole of the UK".
The UK Prime Minister relies on the support of the DUP's 10 MPs for her Commons majority, votes which may become crucial as she attempts to get a deal through Parliament.
A leaked letter from the UK Prime Minister to Mrs Foster and her deputy Nigel Dodds set out Mrs May's approach.
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A Downing Street spokesman said the Mrs May's letter sets out her commitment, which she has been absolutely clear about on any number of occasions, to never accepting any circumstances in which the United Kingdom is divided into two customs territories.
The EU has made concessions on the biggest hurdle in the talks - the Irish border backstop or an emergency fix to ensure the frontier there stays open regardless of how Brexit goes - but there are still differences on the life span of such a solution, a review mechanism or the depth of regulatory alliance under the all-UK backstop, which has been sought by London.
"The government will not agree to anything that brings about a hard border on the island of Ireland", the spokesman said.
Brexit is expected to dominate the agenda of the British Irish Council, which also involves the first ministers of Scotland and Wales, Nicola Sturgeon and Carwyn Jones.
Mrs May says in the letter that the European Union is still pushing for a "backstop to the backstop".
There had been reports that Cabinet will be called for a special session to approve the deal in the coming days.