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".
The party reacted angrily to a letter from Theresa May to DUP leader Arlene Foster, later leaked to the Times.
The Democratic Unionist Party which props up the fragile and divided Conservative government is furious after receiving a letter from May which that refused to rule out a backstop for Northern Ireland alone.
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.
May's letter to the DUP said that she "could not accept there being any circumstances or conditions" for the Northern Ireland-only backstop coming into effect.
Sammy Wilson, the DUP's Brexit spokesperson, accused May of "total betrayal" and ditching the "promises she made" to the party, which has already threatened to withdraw its support for the embattled prime minister. Wright said the government would not accept a Brexit deal which separates Northern Ireland from the rest of the UK.
European Union officials are said to be concerned Mrs May will be unable to convince her Cabinet to sign up to the deal.
Cabinet ministers have presented Theresa May with a detailed plan for a no-deal Brexit amid increasing fears MPs will vote down her deal in the House of Commons.
Ms Foster said: "It appears the prime minister is wedded to the idea of a border down the Irish Sea with Northern Ireland in the European Union single market regulatory regime".
May has asked for a UK-wide backstop to be included in the Withdrawal Agreement instead of the EU's preferred Northern Ireland-only policy.
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.
"There will be product on the table which all will have signed up to that will have involved compromises on all sides", he said.
"I hope and I believe that we can secure that majority in Parliament for the agreement".
Mr Varadkar warned, however, that nothing was guaranteed, adding that any Brexit deal was unlikely to ensure a "clean break".
"The fact that Brexit is happening makes that hard to replicate, but our objective as an Irish Government is to do that to the extent that we can, in order to allow people to travel freely as they have done for so long now, but also to allow trade to function as it does now".
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".
A Cabinet minister said the plan has been developed to ensure the worst elements of a no-deal Brexit could be eased, but they also pointed out it could be beneficial to the United Kingdom in the long run. This would prevent new customs checks between Northern Ireland and Great Britain.