Jeremy Hunt has said there is no possibility of the government backing a customs union with the EU after Brexit.
The health secretary said the UK wanted “frictionless trade” but would “find a different way” to achieve that.
A Conservative MP says she has cross-party support for an amendment calling for the government to form “a” customs union with the EU after Brexit.
Theresa May will make a speech on the UK’s future relationship with the EU next Friday.
And Labour Leader Jeremy Corbyn is set to outline Labour policy on a customs union on Monday, amid reports that his party’s position on it is “evolving”.
If he backs membership of a customs union, it could mean Mrs May faces a Commons showdown over the issue – with pro-European Conservative rebels joining forces with Labour MPs.
Former prime minister Tony Blair said he would welcome a shift in policy from Labour towards a customs union – and if there was an “impasse” in Parliament on the issue, it made the case stronger for a referendum on the final Brexit deal.
But Labour Brexiteer Kate Hoey told the BBC: “A referendum would be ridiculous, less than two years after we had one”.
Conservative MP Anna Soubry says she has cross-party support for an amendment to the Trade Bill, currently going through Parliament, urging the government to pursue as a negotiating objective forming a customs union after Brexit.
A customs union means countries club together and agree to apply the same tariffs to goods from outside the union – but it does not allow members to strike their own trade deals.
Supporters of the UK being in a customs union argue it is vital to protect businesses – but opponents fear it would mean “Brexit in name only” and the UK should make its own arrangements.
Jeremy Hunt told BBC Radio 4’s Today programme a customs union was “one way of getting frictionless trade but it is not the only way”.
The government wanted to agree “frictionless trade by agreement between two sovereign bodies, the United Kingdom and the European Union”, he said.
Asked if there was any possibility of the government coming round to the idea of a customs union with the EU after Brexit, he replied: “No”.
The health secretary was not at the meeting of senior ministers at Chequers on Thursday but said a broad agreement had been reached ahead of a discussion by the whole cabinet and the prime minister’s speech next week.
“You have divergent views on a big issue like Brexit as you would expect – but the central common understanding is that there will be areas and sectors of industry where we agree to align our regulations with European regulations,” he said.
“The automotive industry is perhaps an obvious example because of supply chains that are integrated.
“But it will be on a voluntary basis, we will as a sovereign power have the right to choose to diverge, and what we won’t be doing is accepting changes in rules because the EU unilaterally chooses to make those changes.”
Home Secretary Amber Rudd, who was at Chequers, told the BBC: “We all got behind the prime minister and we’ve agreed the basis for her speech next week and looking forward to it going ahead.”
But pro-EU Labour backbencher Chuka Umunna – an ally of Anna Soubry – warned Theresa May her plan to leave the customs union could be defeated by MPs.
“There is no majority in the House of Commons for us not to participate in the customs union, that is absolutely clear… it’s the reality of the parliamentary arithmetic.”
Of the government, he said: “If they are not going to change their position they are going to lose votes in the House of Commons, it’s a straightforward as that.”
Tony Blair told BBC Radio 4’s The World at One that any “comprehensive customs union” would “mitigate considerably the problems of a hard border between the Republic of Ireland and Northern Ireland” and “protect what we have achieved on the island of Ireland over these past couple of decades”.
‘Cake and eat it’
On Mr Hunt’s comments on frictionless trade without a customs union, he said: “I’m going to be really interested to see that deal because I literally don’t understand what they mean.
“This is ‘cake and eat it’ – it’s not just that Europe has made it clear that it won’t agree that, it’s that the nature of the single market is such is that it has one system of regulation.”
Meanwhile the Times has reported that the prime minister is planning a U-turn over the right of EU citizens who arrive in the UK after Brexit, but during the “transition period”, to remain in the country permanently.
Conservative Brexiteer Jacob Rees-Mogg told the BBC that would be “quite wrong”: “I’d be astonished if Mrs May would make U-turn of that kind; she is a lady of great backbone and for her to kowtow to the European Union is, I think, unconscionable.”