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".
DUP leader, Arlene Foster, expressed outrage at the proposals.
Ms Foster said the DUP will not support a different regulatory regime for Northern Ireland from the rest of the UK.
It has been angered by a letter in which May said EU negotiators were still pushing for Northern Ireland to remain in the single market and customs union if talks collapse.
A leaked letter from the Prime Minister in reply to an earlier message from Mrs Foster and her deputy, Nigel Dodds, set out Mrs May's approach.
Mrs May has also come under fire from her own MPs and Cabinet ministers, who have demanded legal advice on the final terms of the UK's exit to ensure the EU can not keep Britain inside the customs union indefinitely.
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.
"The only thing I would say is that it is very important that we listen to the voice of Northern Ireland in all of this".
The scope of any alignment with Brussels' rules would be limited to what is "strictly necessary" to avoid a hard border.
At issue is the vexing problem of how to avoid border checks between British Northern Ireland and the Republic of Ireland after Brexit enters into force on March 29.
The DUP MP Sammy Wilson heaped further pressure on the government, telling the BBC Radio 4's Today programme that the letter was sent to the party but was put into the public domain by Downing Street.
What does the government say?
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.
Mrs May is said to have told Brexiteers the plan was "not needed yet" but it received a "surprisingly warm" response from Chancellor Philip Hammond, a senior source told Sky News.
Some eurosceptics in her Conservative Party are already threatening to vote against the deal because it could lock Britain into a long-term customs arrangement with the EU.
A UK-wide customs deal would maintain quota-free and tariff-free access to European markets for the British fishing industry and in return the EU wants to keep continued access to United Kingdom waters for its trawlers, the newspaper said.
Speaking at the British-Irish Council meeting, Secretary of State Karen Bradley told the Press Association: "The negotiating teams are working hard to get a good deal that can be taken to the British Parliament".