The warning underscores the travails that May faces in getting any Brexit divorce deal, which London and Brussels say is 95 percent done, approved by both her fractious party and by the Northern Irish lawmakers who keep her in power.
According to The Times of London the letter says the European Union wants a Northern Ireland-specific "backstop to the backstop" and Mrs May tells the DUP she could not accept any circumstances where this could come in to force.
She said the problem with Mrs May's letter is that she has confirmed there will be a Northern Ireland specific backstop and the Chequers plan for the rest of the UK.
Minister for Justice Charlie Flanagan said the issue of British Prime Minister Theresa May writing to the DUP was an internal United Kingdom matter, but said there was a real deep feeling of uncertainty among communities with regard to the border.
DUP Brexit spokesman Sammy Wilson told Sky News that May was guilty of "total betrayal".
A leaked letter from Ms May to Ms Foster and her deputy Nigel Dodds set out the British prime minister's approach.
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The scope of any alignment with Brussels' rules would be limited to what is "strictly necessary" to avoid a hard border.
With less than five months to go until Britain is due to leave the bloc on March 29, both sides remain at odds over how to avoid a hard border in Ireland and are yet to agree a backup plan for the Irish border should a no-deal Brexit occur.
Downing Street says the letter sets out the prime minister's commitment "to never accepting any circumstances in which the United Kingdom is divided into two customs territories".
"It must be built on to ensure our rights are protected and translated into legally operable and binding text as part of any withdrawal agreement".
Speaking at the British-Irish Council summit on the Isle of Man, Mr Varadkar said he was hopeful a Brexit deal could be done by the end of the year but it would not amount to a "clean break" as talks would have to continue. She said: "I will make no bones about the fact that the Brexit experience has exposed real weaknesses within the devolution settlement that will require them to be worked through and resolved along the way".
Downing Street has played down suggestions that a Brexit deal is imminent, after European Council president Donald Tusk appeared to indicate a breakthrough could come within the next week.
Cabinet ministers have been on stand-by for an emergency session to review any potential deal, but the source said that "nothing is going to happen" in terms of a meeting over the weekend.