First Minister Nicola Sturgeon warned that without such an arrangement being in place, Scotland would be left competing for investment with a country which was “effectively” still part of the trading bloc.
Her message, echoed by SNP Westminster leader Ian Blackford, came as Theresa May prepared to face her Cabinet in a showdown meeting over the draft Brexit deal.
Ms Sturgeon said on Twitter that the “PM’s approach would take Scotland out of the single market (despite our 62% remain vote) but leave us competing for investment with Northern Ireland that is effectively still in it”.
She later told ITV Border: “This from what I know of it is not a good deal for Scotland.
“It would take Scotland out of the single market which would be bad enough in and of itself, but it would do so while leaving us competing for investment and jobs with Northern Ireland that would effectively be staying in the single market. That would be the worst of all possible worlds for Scotland.
“So what I think is important now is that we don’t allow ourselves to fall for the Prime Minister’s spin that a bad deal has to be accepted because the only alternative is no deal.
“It does not mean that no deal is inevitable if this deal is rejected by the House of Commons. Instead what that opens the possibility for is getting better options like full single market customs union membership back on the table. We must be striving for what is right for the country. My job is to strive for what is right for Scotland, not accepting bad deals.
“It’s ironic, is it not, that for the last two years the Prime Minister has told us that a no deal is better than a bad deal and now she’s arguing that we’ve got to accept a bad deal for fear of no deal.
“That is a false choice.”
Her remarks came as Democratic Unionist Party chief whip Sir Jeffrey Donaldson warned the arrangements for Northern Ireland could lead to the break-up of the United Kingdom.
He claimed the deal would in the “long term” leave Northern Ireland closely aligned with the EU and could increase support for Scottish independence.
“It’s about the constitutional and economic integrity of the UK, that is fundamental for us,” he told BBC Radio, adding that the “DUP does not stand alone on this, we have many friends within the Conservative Party and indeed in some other parties, who believe this deal has the potential to lead to the break-up of the UK”.
Scottish Conservative leader Ruth Davidson, who is currently on maternity leave after the birth of her son, and Scottish Secretary David Mundell have both warned they could quit their roles if Brexit threatens to “undermine the integrity” of the UK.
Mr Blackford, an SNP MP, said the two Tories had “got questions to ask”, but added: “I’ll leave that to them.”
For her part, Ms Sturgeon said: “I’ll leave David Mundell to worry about his own contortions. The Scottish Conservative party in general and David Mundell in particular have got themselves into all sorts of strange postions over this. They seem more interested in standing up for the DUP than standing up for Scotland which I think is a strange position for the Secretary of State for Scotland to be in.
“My job is to stand up for Scotland. “Scotland voted to remain overwhelmingly. Not withstanding that I have argued consistently for a compromise position of single market and customs union membership.
“What I don’t think anybody with Scotland’ best interests at heart could accept is a situation where Scotland is not just out of the single market but competing with Northern Ireland which would still be in the single market. That would be devastating for jobs and investment in Scotland and not something I think that anybody should be prepared to accept.”
Mr Blackford stressed that Scotland should be treated the same as Northern Ireland.
“As far as we understand things this morning it looks as if it’s actually going to a be a different deal for Northern Ireland,” he told BBC Radio Scotland’s Good Morning Scotland.
“If it’s permissible for Northern Ireland to stay in the single market as part of the backstop - and I welcome that for the people of Northern Ireland - then of course Scotland should be given the same opportunity.”
He continued: “Nicola Sturgeon, our First Minister, has been very clear ever since the Brexit vote that we need to respect the votes of the people of Scotland and as a very minimum that means staying in the single market and the customs union, these are our red lines.”
Mr Blackford insisted it was “absolutely right” that the Brexit deal should enshrine the Good Friday agreement, which ended years of conflict in Northern Ireland, and ensure there is no return to a hard border with the Irish Republic.
But he went on: “The key point is that if there is to be a differentiation for Northern Ireland there is no reason why the circumstances in Scotland can also not be respected.
“We need to stay in the single market and the customs union, and the Government in London must listen to us on that.”
Scottish Conservative chief whip Maurice Golden said: “The UK Government and the European Union have negotiated a plan to avoid a hard border between Ireland and Northern Ireland.
“This shouldn’t be about creating a new border between us and England.
“Nicola Sturgeon has spent months talking up the merits of the EU single market.
“She needs to explain why it is in Scotland’s interests to fracture the UK internal market, as she seems to want to do.
“The rest of the United Kingdom is four times as important in terms of trade to Scotland than the entire European Union.
“It simply does not make sense to create a border at Berwick by creating different trading rules between us and England.”