But she stresses that she would never allow a divide between Northern Ireland and Britain to "come into force".
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. To say, "because I've been incompetent in the negotiation so far I'm going to have to agree to a final backstop that could undermine the integrity of the United Kingdom, but go on to say, trust me because I'll be more competent in my future negotiation than I have in the ones to date", is a line that lacks credibility'.
However, Brussels has demanded an insurance policy to avoid a hard border in Ireland.
Mrs May has reportedly written to the Northern Irish party saying that the European Union is still pushing for a so-called "backstop to the backstop".
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.
Mrs Foster said: "The Prime Minister's letter raises 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".
In the leaked letter, the prime minister also told the DUP she "would not accept being kept in a backstop arrangement indefinitely", while it would be "totally unacceptable" for a time limit to the UK-wide backstop arrangement to then simply result in the Northern Ireland-only solution being adopted.
Taoiseach Leo Varadkar, Mrs May's effective deputy prime minister David Lidington and Northern Ireland Secretary Karen Bradley will attend a summit on the Isle of Man on Friday.
'The government will not agree anything that brings about a hard border on the island of Ireland'.
However, in the leaked letter, it is revealed Mrs May admitted to the DUP that the European Union is still proposing a Northern Ireland-only backstop arrangement alongside a UK-wide solution.
There had been reports that Cabinet will be called for a special session to approve the deal in the coming days.