The DUP has said the European Union "must compromise" on the backstop arrangements or we are facing a no-deal Brexit.
Speaking on RTÉ's Morning Ireland, Jeffrey Donaldson said that while the DUP wanted to leave with a deal, it would not join parliamentary efforts to prevent Brexit from happening and would not take no-deal "off the table".
"We know that is the strongest lever the UK government has in its negotiations with Brussels," Mr Donaldson said.
However, he stressed the DUP wanted to see a deal and he hopes that Brussels would compromise on a withdrawal agreement.
He said a no-deal scenario can be prevented, not just by the House of Commons but by the Irish Government "being pragmatic and prepared to compromise".
Mr Donaldson said there was a great desire in Dublin "to ignore the realities coming down the track" and not to focus on the impact a no-deal Brexit would have on the Republic, but on Northern Ireland.
He said there was a massive impact on the Republic and so it is vital the Irish Government, the EU and the UK work together to get a withdrawal agreement in place.
Mr Donaldson said he does not accept the doom-laden forecasts, but acknowledges the difficulties a no-deal situation would bring to the island of Ireland.
Minister for Business, Enterprise and Innovation Heather Humphreys said the business community north and south in Ireland wants to hear the new British Prime Minister's plans for the border post-Brexit as soon as possible.
She said solutions have to be found to deal with cross-border trade after Brexit, as the Irish position remains unchanged about the backstop and there is "no point in Boris Johnson saying 'we can sort it out', without putting meat on the bone".
Speaking on the same programme, she said businesses want to know what to expect after October and it was proving "very difficult" to prepare with the uncertainty surrounding the issue.
"That's why the backstop is there", she added.
Minister Humphreys said it was not easy trying to unravel years of business practices along the border and from east to west.
She said it was incumbent on the UK, the EU and Ireland to protect the Good Friday Agreement as an international agreement and as someone who had lived through the Troubles, she could not "countenance going back to a border and all it entails".
She said she accepted that checks on goods would be required to protect the integrity of the EU single market and there was a very intricate supply chain between the Republic and Northern Ireland where components and goods criss-cross the border seamlessly.
To unravel that and put in checks was going to be very difficult, she added.
Fianna Fáil's Brexit spokesperson said she is quite concerned with the new make-up of Mr Johnson's cabinet.
Lisa Chambers said that Mr Johnson has effectively assembled a wartime cabinet, adding that perhaps he was reading too many Churchill biographies.
Speaking on RTÉ's Today with Miriam, she said the cabinet should "take a dose of reality and try and get a deal over the line".
Ms Chambers added that the suggestion that the UK can leave on a "managed no deal" is not reality and the choices are leave with a deal or crash out.
She said she believed Mr Johnson is engaging in a "game of chicken" and did not believe there would be any change until October, when the UK was heading towards a cliff edge.
