Theresa May is under massive cross-party pressure to grant MPs a vote on any decision to leave or limit UK involvement in the European single market, amid growing outrage at the prospect that parliament could be bypassed over the biggest economic decision in decades.

Tory MPs joined forces with former leaders of Labour and the Liberal Democrats, the SNP and Greens to insist that parliament have a say and a vote, pointing out that, while the British people had backed leaving the EU, they had not chosen to leave the biggest trading market in the western world.

Former Labour leader Ed Miliband held discussions with pro-EU Tory MPs on Saturday, and was said to be considering tabling an urgent question in the Commons, demanding that May appear before parliament to explain its future role in Brexit decisions, when MPs return on Monday.

The SNP and pro-EU Tory MPs Nicky Morgan and Anna Soubry were also considering tabling questions, while former Lib Dem leader Nick Clegg, now the party’s Brexit spokesman, said it would be appalling if detailed terms of Brexit, including the UK’s future relations with the single market, were not voted on by MPs.

Miliband told the Observer: “Having claimed that the referendum was about returning sovereignty to Britain, it would be a complete outrage if May were to determine the terms of Brexit without a mandate from parliament.

“There is no mandate for hard Brexit, and I don’t believe there is a majority in parliament for [it] either. Given the importance of these decisions for the UK economy … it has to be a matter for MPs.”

Clegg said: “My great worry is that while there will be a vote on repealing the 1972 European Communities Act, which is about the decision to leave the EU, it will be left to the executive alone to decide the terms of Brexit. That would not be remotely acceptable.”

The extraordinary cross-party alliance has formed since ministers made a succession of speeches at the Tory party conference in Birmingham last week, which left the clear impression they would press ahead with a hard Brexit. The suggestion was that this would mean leaving the single market if that was the cost of regaining control of the UK’s borders.

Extraordinary cross-party alliance has formed since Tory party conference in Birmingham. Photograph: Stefan Rousseau/PA

Concern has also grown after business leaders, led by the Confederation of British Industry and EEF (formerly the Engineering Employers Federation), wrote an open letter to the prime minister, warning that a hard Brexit that failed to secure a close relationship with the single market should be ruled out “under any circumstances”.

The letter says the terms on which the UK leaves the EU will be critical to the UK economy, jobs and future investment, while resorting to World Trade Organisation (WTO) rules would leave 90% of UK trading in goods subject to new tariffs. This would mean an additional 20% in extra costs for the UK’s food and drink industry and 10% for car producers.

“We respect the result of the referendum, but the government must make sure that the terms of the deal to leave ensure stability, prosperity and improved living standards,” the business groups say.

MPs demanding a vote point out that the 2015 Tory manifesto made explicit that, while there would be an in/out referendum, membership of the single market would still be in the country’s best economic interests. The manifesto said “We say: ‘Yes to the single market’”, and that the party would safeguard the country’s interests within it.

May has only said there will be a vote on a repeal bill to remove the 1972 act from the statute book, and that parliament will be kept informed when possible on negotiations on the detailed terms. But there is no prospect of a vote on what form Brexit will take, nor the terms.

Ministers, including chancellor Philip Hammond, have been at pains to stress that the government will seek the best possible “access” to the single market.

Labour’s former shadow business secretary, Chuka Umunna, said: “We conclude from the Tory conference that they are waving the white flag on remaining in the single market before negotiations have even begun. If there is any move away from it, there should be consultation in parliament and a vote.”

Norway is a member of the single market but not a member of the EU and has to pay into its budget and accept free movement rules.

Green party co-leader Caroline Lucas said: “The assumption that those who voted to leave were also signalling their wish to withdraw from the single market is unfounded. And for a government whose own manifesto commits it to continued membership of the single market, to now seek to overturn that and reinterpret the referendum result, while denying people a say through a vote in parliament, is a constitutional outrage.

“Of course, the referendum result must be honoured – but the vote was to leave the EU, not to inflict the greatest possible damage to our economy and tear apart community cohesion. ”

David Cameron’s former policy guru condemned May and her government for the direction she is taking the country in over Brexit. Steve Hilton, who broke with the former PM to back Brexit, was scathing about home secretary Amber Rudd’s suggestion to make firms list the number of foreign workers they employ. Ministers might as well announce that “foreign workers will be tattooed with numbers on their forearms”, he wrote in the Sunday Times.

At the Tory conference, May said a Great Repeal bill would be put forward in the Queen’s speech next spring and would be voted on in the next session of parliament, coming into force on the day of exit. Ministers insist they cannot give a “running commentary” on the details of negotiations.

A No 10 source said: “As the prime minister said in her conference speech, we’re getting on with the job of delivering Brexit, and we’re going to make a success of it. Building relationships is absolutely essential in this business and putting in the hard yards in this way means Theresa will be one of the best travelled PMs by the time she meets everyone at her first European council meeting next month.”

A source at the Department for Exiting the European Union said: “Parliament voted by six to one to give people the final say on EU membership in the referendum held on 23 June. While there can therefore be no attempt to keep Britain in the EU by the back door, we have been clear that parliament will be consulted and engaged throughout the process of exit.

“We will introduce a Great Repeal bill to end the authority of EU law and we have been clear that we will follow all the constitutional and legal precedents that apply to any treaty on a new relationship with the EU.”