At least 10 Cabinet ministers could back a Norway-style Brexit if Theresa May’s deal is rejected, one of the MPs leading a cross-party campaign has claimed.

Under the Norway-plus plan, the UK would remain in the single market and customs union, which would remove the need for the Irish backstop provision and protect trade links with the EU, but it would fail to deliver on key Brexiteer promises over free movement and sovereignty.

If the Prime Minister’s Brexit plan is defeated next week, her Cabinet colleagues could argue for the Norway-plus option, according to an MP involved in cross-party efforts behind the plan.

Labour’s Stephen Kinnock said: “The country is crying out for us to put our narrow tribalism to one side.

“We understand that there are at least 10 Cabinet ministers who are supporting this arrangement.”

He told BBC Radio 4’s Today programme: “What we need the Prime Minister to do, when she loses the vote on Tuesday, is to go on to the steps of Downing Street and make a very clear statement that we must pivot now to Norway-plus.”

A Norwegian-style compromise is opposed by many Eurosceptics, who believe it will amount to “Brexit in name only”, and by former Remainers who are pushing for a second referendum.

There is also resistance within Europe about the possibility of an economy the size of the UK joining the European Free Trade Association (EFTA) and European Economic Area (EEA) along with Norway, Iceland and Liechtenstein.

Norwegian MP Heidi Nordby Lunde told The Guardian: “Really, the Norwegian option is not an option.

“We have been telling you this for one and a half years since the referendum and how this works, so I am surprised that after all these years it is still part of the grown-up debate in the UK.

“You just expect us to give you an invitation rather than consider whether Norway would want to give you such an invitation. It might be in your interest to use our agreement, but it would not be in our interest.”

The People’s Vote campaign, which wants a second referendum, published a dossier condemning the Norway-plus proposal.

In a joint foreword, Labour former foreign secretary David Miliband and Tory ex-minister Jo Johnson – who quit his Government role in November over Brexit – said: “Norway-plus would represent a long-term commitment to pay to benefit from the European Union’s regulatory structures while choosing to be outside it.”

They added that while rejoining EFTA may limit the economic damage it would represent a “significant loss of power” and influence.

“Both sides would be entitled to ask, if the past two and a half years of upheaval and tortuous negotiation culminated in a Norway-plus deal, what was the point of all that?”

Mr Johnson said the arrangement would turn the UK into a “nation of lobbyists” seeking to influence rules set in Brussels over which it had no say.

But Tory ex-minister Nick Boles said: “Norway-plus is a compromise that has broad appeal to the pragmatic middle.

“It delivers a softish Brexit with a deal that preserves membership of the single market and keeps the union of the UK intact.”