"That is why it is critical that the provision for a UK-EU joint customs territory is legally binding in the Withdrawal Agreement itself, so that no "backstop to a backstop" is required".
Mrs May insists in her letter to Mrs Foster that she would never allow this second backstop to be activated.
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.
Brussels has said the United Kingdom will not be able to dissolve the backstop without agreement from the EU.
"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".
Mrs Foster told the newspaper: "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. That's a decision in exceptional circumstances for the Prime Minister".
The EU has made concessions on the biggest hurdle in the talks - the Irish border backstop or an emergency fix to ensure the frontier there stays open regardless of how Brexit goes - but there are still differences on the life span of such a solution, a review mechanism or the depth of regulatory alliance under the all-UK backstop, which has been sought by London.
"The government will not agree to anything that brings about a hard border on the island of Ireland", the spokesman said.
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.
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.
A potential sticking point could be demands for European Union fishing fleets to be given continued access to British coastal waters as the price for agreeing to Mrs May's UK-wide backstop, the Daily Telegraph reported.
It came as Mrs May embarked on a diplomatic tour of Europe as she raced to get a Brexit deal ready to present to her Cabinet as soon as Monday.