The DUP leader said proposals for a “backstop” arrangement – effectively keeping Northern Ireland in the single market and custom union to avoid a hard border on the island of Ireland – would lead to its “permanent annexation” from the UK.
It comes as the Times newspaper reports the PM is contemplating ending the Conservative-DUP agreement in order get her Brexit deal through Parliament.
Writing in the Belfast Telegraph, Mrs Foster said the DUP wants Mrs May to “stand by her principles and instincts rather than accepting a dodgy deal foisted on her by others”.
“I fully appreciate the risks of a ‘no deal’ but the dangers of a bad deal are worse,” she wrote.
“If true to her principles Mrs May would not and should not choose the path to effectively cut Northern Ireland adrift.”
She added: “This backstop arrangement would not be temporary. It would be the permanent annexation of Northern Ireland away from the rest of the United Kingdom and forever leave us subject to rules made in a place where we have no say.”
Mrs Foster said the PM should learn from Margaret Thatcher who “deeply regretted” being persuaded to sign the 1985 Anglo-Irish Agreement, which gave the Irish government a formal say in Northern Ireland affairs for the first time.
She also denied the DUP’s recent actions – including a threat to vote down the budget – represented the party “flexing muscle”.
The DUP’s 10 MPs prop up the Tory government, which has no majority.
Mrs May travels to Brussels on Wednesday for a meeting of the European Council, consisting of the leaders of the bloc’s 28 member states.
Mrs Foster has made clear that any Brexit deal in which Northern Ireland is treated separately to the rest of the UK, or faces extra checks along the border with Ireland, would be unacceptable to her party and that this is a “blood red line”.
At a Leave Means Leave rally in Torquay, Jacob Rees-Mogg said that if there was a “punishment Brexit” and chaos at Dover, it would demonstrate that it had been the correct decision to leave the EU.
Laughing off a question about whether he wanted to become prime minister, he told Sky News: “There cannot be a separate arrangement for Northern Ireland.
“Northern Ireland is as much a part of my country as Somerset is – and the two parts of the country must be treated equally and fairly and to cast Northern Ireland off into a colony of the European Union would be entirely improper.”
Writing in the Belfast News Letter, Boris Johnson also called on Downing Street to scrap the Irish backstop.
The former foreign secretary said: “The fatal error was not to challenge the EU’s position that the only way of avoiding a hard border on the island of Ireland – an objective we all share – is for Northern Ireland to have the same regulations for trade as Ireland and the rest of the EU.
“The government well understands how weak the backstop makes its future negotiating hand… if we let this go it will be the greatest national humiliation since Suez.”