The DUP leader Arlene Foster says she does not accept there needs to be a new Conservative leader for her party's desired amendments to be made to the Brexit deal.
Mrs Foster said Prime Minister Theresa May was aware of the DUP's difficulties with the agreement.
The pair met for about 45 minutes at Westminster earlier on Wednesday.
Her comments came ahead of a no-confidence vote in the Conservative leader.
The DUP's 10 MPs prop up Mrs May's government under a confidence-and-supply agreement signed in June 2017.
The Conservative leader and prime minister had been planning to discuss the Brexit negotiations with the taoiseach (Irish prime minister) on Wednesday.
However the trip to Dublin had to be cancelled after 48 Conservative MPs triggered a vote of no-confidence in Mrs May's leadership.
It comes after a turbulent week in Westminster.
Speaking after the vote, the DUP deputy leader Nigel Dodds said that there would be a "heck of a battle" with the EU and Irish government over the arrangements for the Irish border backstop.
Mr Dodds, who also met the prime minister along with his party leader, said that Mrs May had "accepted" during their conversations on Wednesday that how "the process had been handled up to now... left a lot to be desired".
However, despite their reservations over the existing deal, Mr Dodds said that the DUP would support the government in the event that it faced a no-confidence vote.
Majority of MPs back May
On Monday, Mrs May announced that a vote in the House of Commons on her Brexit deal, planned for Tuesday, would be postponed as she sought to get assurances on the backstop.
A majority of Conservative MPs have publicly backed Mrs May, but it is a secret ballot.
It is understood that Mrs May has told backbench MPs she will not lead the party into the next scheduled election in 2022, but is seeking support now to enable her to get her Brexit deal through.
Mrs Foster told BBC News NI that she does not believe the kind of radical alterations her party wants to the government's Brexit policy require a new Conservative leader.
The DUP leader said Mrs May understands what is needed regarding the Irish border backstop and added that it is not just the DUP that has an issue with the Northern Ireland protocol.
Mrs Foster said: "The prime minister has known our position. We have been consistent which is why it is so frustrating that our warnings about the backstop have not been heeded.
"The DUP wants a sensible deal which our MPs can support in the House. Unionism in Northern Ireland and across the House of Commons has rightly stood against this Withdrawal Agreement.
"It should be utterly unacceptable to any unionist. For Northern Ireland traders to be expected to treat GB as a third country is ridiculous and was never going to receive support in Parliament."
Mrs Foster said the prime minister was in a mood to listen to the DUP's concerns, and she put forward alternative ways to deal with the issue of a hard border.
During Northern Ireland questions, the Secretary of State Karen Bradley told MPs the idea of using technology to manage the border after Brexit was flawed.
Assembly promise sought
Mrs Bradley argued that "at this time there is no technology which deals with the issue of the border in a way that respects the rights of the people of Northern Ireland and also respects the Belfast Agreement and the way it operates".
However the DUP leader said that it is for the prime minister to decide on such matters.
Mrs Foster also reiterated that a promise that a future Northern Ireland Assembly and Executive would have a lock on whether trade regulations in Northern Ireland diverge from Great Britain should have been included within the EU Withdrawal Agreement.
Questioned about forecasts that more than 170 Conservative MPs have pledged support for Mrs May, the DUP leader refused to speculate on the result of the vote, insisting that is a matter for the Conservative Party.