Post-Brexit rules mean checks are required on animal-based products entering the UK from the EU.
And because NI continues to follow EU customs rules, the regulations also apply to goods moving between Great Britain and NI.
A 'Swiss-style' trading arrangement would see some checks on animal-based products loosened, therefore mitigating trade disruption.
But the DUP's economy minister Diane Dodds on Tuesday ruled out such an arrangement with the bloc, saying it would require the whole of the UK to "slavishly" follow EU rules "in every respect".
Her position was criticised by politicians from across the spectrum in Northern Ireland, including the Ulster Unionist Party leader and South Antrim MLA Steve Aiken OBE.
Mr Aiken, who is calling for the protocol to be scrapped entirely, said a Swiss deal “should be welcomed”.
Alliance Party deputy leader and North Down MP Stephen Farry accused the DUP of failing to protect business interests.
"There is a consensus across a range of political parties and amongst [the] business community that this is an area where progress could and should be made," Mr Farry said of the proposal.
"The DUP are fundamentally reading this wrong, and are not defending economic interests of NI."
Matthew O'Toole, an SDLP MLA who put the proposal to Ms Dodds in Stormont on Tuesday, accused the minister of rejecting it "on the basis of Brexit ideology".
Appearing on the BBC’s Good Morning Ulster show on Wednesday morning, Ms Dodds's stance had shifted somewhat following the criticism.
She suggested her comments had only been reflecting Conservative Party policy, adding she would support a Swiss-style deal if the government pursued such an arrangement.
"I'm not against anything that will bring short-term relief," she said. "I'm not against a Swiss-style deal in and of itself. If they support that, that is fine”.
Ms Dodds added: "What I was reflecting yesterday was that the whole of the UK will have to sign up to that, so the UK will have to follow EU rules."
Under the terms of the Brexit agreement, NI is continuing to follow the EU's single market and customs rules to protect the Good Friday agreement and prevent a hard border on the island of Ireland.
The new rules, essentially putting a border down the Irish Sea, mean that products coming into NI from the rest of the UK are subject to checks, resulting in some trade disruption.
The most onerous checks are on animal-based products because the EU and the UK follow different rules on food standards.
A Swiss-style arrangement would see greater alignment between the UK and the EU on veterinary, sanitary and phytosanitary checks.
This would allow animal products to move more freely between the EU and the UK and therefore into NI.
The secretary of state for environment, food and rural affairs, George Eustice, has previously said he is not ruling out a Swiss-style deal.
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