A senior source said: "It does not feel to me like things are going to move today".
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.
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".
While the Prime Minister said she would not allow the second backstop to "come into force", her letter failed to rule out it being a part of the finalised agreement.
He said the two sides were discussing how to avoid a hard border between Ireland and Northern Ireland.
The DUP has seized on a particular paragraph- which has been seen by the Times - in which May said she could not accept circumstances or conditions that could break up the United Kingdom customs territory to come in to force.
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.
LONDON-The Northern Irish party which props up Prime Minister Theresa May's government, cast her Brexit negotiation as a betrayal and cautioned it could not support a deal that divided the United Kingdom.
The Prime Minister relies on the support of the DUP's 10 MPs for her Commons majority, votes which will be crucial as she attempts to get a deal through Parliament.
Less than five months before Britain is due to leave the European Union on March 29, negotiators are still haggling over a backup plan for the land border between British-ruled Northern Ireland and European Union member Ireland should they fail to clinch a deal.
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.
"And I think if we do that, if we listen to the voice of Northern Ireland as a whole that will help us to come to an agreement".
Speaking this afternoon in the Isle of Man, Leo Varadkar says a Brexit agreement is possible in the near future.
"There is no clean break here, Brexit is going to go on for a very long time".
"People will need to ask themselves what is it that is going to be in the best interests of those who sent them to Westminster to represent them, to ensure that we maintain living standards and investment and prosperity and employment in our country".