The DUP has stepped up the pressure on Theresa May over its demands for Brexit, by briefing that it could vote against this month’s budget if the prime minister breaches the party’s red lines on the issue at next week’s EU summit.

The threat – which could in theory topple May, or even bring down the government – has not been made publicly, but was briefed to the BBC and Sky News by sources in the Northern Irish party.

According to the reports, if the DUP feels May has given away too much at next week’s crunch meeting in Brussels, the party could withdraw the support of its 10 MPs for the vote on the budget, thus risking it being defeated.

Traditionally, budget votes were seen as issues of confidence in a government, triggering a general election if one was lost. This is no longer the case under the fixed-term parliament act, but such a defeat could fatally undermine May’s position and would bring calls for a vote on calling an election.

The DUP, which has had a so-called confidence and supply agreement to support May’s government since last year’s election, has expressed regular alarm at the prospect of any Brexit deal avoiding the possibility of a hard Irish border by instead making Northern Ireland’s regulatory regime more closely allied to the EU, creating a de facto customs border in the Irish Sea.

Dutch and DUP strike double blow to May's Brexit plans Read more

At the Conservative conference last week, Nigel Dodds, the leader of the DUP in the House of Commons, said the party would vote against May if she returned from Brussels with a deal that involved new checks on goods coming into Northern Ireland from Britain. “We will vote against it. We will vote for our red lines,” he said.

The DUP leader, Arlene Foster, had earlier said the party’s confidence and supply arrangement with the Conservatives was “party to party” and not with May herself – prompting speculation the DUP might be willing to force May from No 10.

Foster reiterated her party’s stance during meetings in Brussels on Tuesday. “I am the leader of the Democratic Unionist party. The clue is in the title,” Foster said after a meeting with the EU’s chief negotiator, Michel Barnier.

“I am a unionist, I believe in the union of the United Kingdom, all four elements of the United Kingdom. We do not want Northern Ireland going off in a different direction from the rest of the UK.”

The Treasury minister Mel Stride said: “The prime minister’s position has always been crystal clear on this issue of no border down the Irish Sea. We have said it many, many times.”

The DUP has vowed to oppose any fresh barriers to trade between Great Britain and Northern Ireland.

Stride told BBC Radio 4’s World at One: “I’m not going to get into the specifics of what may or may not be going on in the negotiations at the moment, other than to say that I am extremely confident from all that I know and have seen, and all the discussions that I have had, that there will be a very, very firm position taken on this.

“The prime minister made it very clear no UK prime minister is ever going to put him or herself in the position where they start to unpick the economic and sovereign integrity of the UK.”

May has also faced the threat of parliamentary guerrilla tactics from Brexiters within her own party who are understood to be seeking an opportunity to stage a show of strength in protest at her Brexit plans.

The former Brexit minister Steve Baker has claimed 40 of his colleagues were determined to vote against the Chequers proposal, whatever the consequences.

Failing that, some members of the ERG have suggested they could vote down parts of the budget, although the tactic would be particularly controversial as it could precipitate the fall of the government.