Theresa May’s de facto deputy, David Lidington, has insisted the government remains confident it can get its Brexit deal through parliament, despite the Democratic Unionist party warning it is prepared to vote it down.

Speaking in the Isle of Man, where he was attending a meeting of the British-Irish Council, Lidington said he believed a “new dynamic” would emerge once MPs saw the full text of the proposed agreement.

He said: “I hope and I believe that we can secure that majority in parliament.”

Lidington was speaking after the DUP’s Brexit spokesman, Sammy Wilson, said his party was prepared to vote against May’s deal.

The Northern Irish party, upon which May upon for her Commons majority, had reacted angrily to suggestions in a letter from the prime minister, which had been leaked to the Times, that Northern Ireland could have a different regulatory regime to the rest of the UK if the Irish backstop comes into force.

Wilson said Conservative MPs also had concerns about other aspects of the mooted deal. “If she continues down the road of bringing something forward which is unacceptable to a large part of her own party and ourselves, then I think the inevitable consequence is that it will be voted down in the House of Commons,” Wilson told Sky News.

The DUP accused May of breaking a promise that she would never sign up to a deal that treated Northern Ireland differently from the rest of the UK.

The party has seized on a particular paragraph of the letter in which May said she could not allow circumstances or conditions that could break up the UK customs territory to “come into force”.

The EU has insisted on a Northern Ireland-only “backstop to the backstop” in case negotiations on a wider UK approach break down. Any version of the backstop would apply unless and until a wider UK-EU deal on the future relationship solved the issue of how to avoid a hard border in Ireland.

David Lidington: ‘The prime minister has always been clear that we won’t accept something that involves carving out Northern Ireland from the rest of the UK.’ Photograph: Simon Dawson/Reuters

Wilson said: “She is now contemplating signing up to a legal agreement which, regardless to her aspirations, would be binding on the government of the UK. Secondly, it would be a legal agreement which the government of the UK could not walk away from – that could only be broken if the government in the UK and the EU agreed to it being changed.”

The DUP leader, Arlene Foster, tweeted on Friday: “The PM’s letter raises alarm bells for those who value the integrity of our precious union & for those who want a proper Brexit for the whole UK. From her letter, it appears the PM is wedded to the idea of a border down the Irish Sea with NI in the EU SM [single market] regulatory regime.”

Lidington sought to soothe those concerns on Friday. He said: “The prime minister has always been clear that we won’t accept something that involves carving out Northern Ireland from the rest of the UK.”

The letter is seen by some observers, as well as the DUP, as part of a laying of the groundwork by the prime minister for a showdown with the party over checks in British ports, or factories in Northern Ireland or Great Britain.

Under the EU proposals, UK officials would be “competent authorities” to conduct the checks, but it is something the DUP has said will cross its red line.

If May can win the final backing of her cabinet for her negotiating stance, she hopes to strike a deal with the EU in the coming days.

That would include both a withdrawal agreement, governing the terms of Britain’s exit, and a political declaration about the future trading relationship the two sides hope to negotiate after Brexit. Both documents would then be published.

“I think that if we get the withdrawal agreement, and the accompanying political declaration, which I hope that we can secure in the weeks to come, that will create a new dynamic,” Lidington said.

He continued: “It will no longer be a question of some hypothetical outcome, to which anybody might wish to add or subtract their own political preferences, but a pair of documents that will have been agreed and endorsed by the government of the United Kingdom, and by the 27 other governments of the member-states of the EU, led by politicians of left, right and centre.

“There will be a product on the table, which all have signed up to – which will have involved compromises, give and take on all sides.”

The Irish prime minister, Leo Varadkar, speaking alongside Lidington, said as well as the DUP it was important to listen to “Northern Ireland businesses, and Northern Ireland farmers, and people who live in Northern Ireland”.

He said he was hopeful a deal could be struck within the next couple of weeks, but meanwhile “the less said the better about the detail”.