The DUP wants to "get Brexit done" but it must be for the whole of the United Kingdom, it's spokesman on leaving the European Union has said.
On Sunday, DUP East Antrim MP Sammy Wilson said the party position has been "clear and consistent".
He said the 10 DUP MPs had supported the Letwin amendment in the House of Commons on Saturday to delay voting on British prime minister Boris Johnson's latest Brexit agreement with Europe "as the only avenue available to properly scrutinise" it and "attempt to secure changes that could address some of the concerns we have".
“The DUP does not seek a second referendum; merely implementation of the first,” he said.
In March DUP deputy leader and Westminster leader Nigel Dodds said he would "rather stay in the EU ... rather than risk Northern Ireland's position" in the UK.
Also on Sunday DUP South Belfast MP Emma Little Pengelly told the BBC it was "regrettable" the party, which is in a confidence and supply agreement with the Conservatives at Westminster, had been "presented with a deal we cannot support".
She said DUP votes had been critical on Super Saturday in the House of Commons and it is clear Mr Johnson “does need the DUP”.
Ms Little Pengelly reiterated the DUP desire to see the Brexit leave vote implemented and claimed Mr Johnson’s proposals around the customs union and single market for the North “put significant barriers to our main market”.
“The union is our priority,” she said.
She added: “The entire of the UK must leave in a way that is good and sensible for all parts.”
All unionist parties in the North are unhappy with the new proposals Mr Johnson agreed with the EU last week.
UUP peer Lord Reg Empey says the DUP "have proved to be out of their depth".
He argued their electoral opponents “opened the floodgates to the Irish Sea border having broken their own promises and red lines”.
“It is the DUP who have led us to this situation,” he said.
Outgoing UUP leader Robin Swann is calling for cross-party cooperation at Westminster "to ensure that the wrongs that would have been foisted on the people of Northern Ireland through Boris Johnson's deal are corrected".
Mr Swann is expected to be replaced shortly by South Antrim MLA Steve Aiken who is backing a remain position in light of the threat he believes the latest agreement poses to the North’s union with Britain.
PUP leader Billy Hutchinson has said "regulatory checks and customs controls between Great Britain and Northern Ireland fly in the face of the Belfast Agreement".
He is concerned about the damage the deal could have on the North’s constitutional position in the UK and called again for unionists from all backgrounds to “put country before party” and join together in discussion.
Meanwhile, an unnamed loyalist source has been briefing journalists in recent weeks that civil unrest will take place if a bad deal for the North goes ahead.
However, PSNI chief constable Simon Byrne recently said officers anticipate over time a change in attacks from dissident republicans, depending on how Brexit plays out.
He said there could be “some backlash” from loyalists if jobs and agriculture are put at risk but “at the moment our assessment is a slow time change”.
DUP Westminster leader Nigel Dodds told the Commons on Saturday: “Weariness in this House over Brexit should not be an excuse for weakness on Brexit or weakness on the union.”
He said there must be “Brexit for the whole of the United Kingdom”, leaving the single market and customs union as one.
Mr Dodds raised concerns over Northern Ireland’s involvement in the VAT regime and the single market “without any consent up front” before claiming: “It drives a coach and horses through the Belfast Agreement by altering the cross-community consent mechanism.”
Mr Dodds then alluded to previous warnings from Mr Johnson about how no British PM could agree to such terms, adding: “Will he now abide by that and please reconsider the fact that we must leave as one nation together?”
Replying Mr Johnson said together he and the DUP secured changes on the customs union before defending the measures in the deal for Northern Ireland. He said: “In all frankness I do think it a pity that it is thought necessary for one side or the other in the debate in Northern Ireland to have a veto on those arrangements.
“Because after all, I must be very frank about this, the people of this country have taken a great decision embracing the entire four nations of this country by a simple majority vote that went 52-48, which we’re honouring now. I think that principle should be applied elsewhere and I see no reason why it should not be applied in Northern Ireland, and it is in full compatibility with the Good Friday Agreement.”