Pro-remain MPs predicted on Saturday that they were gaining sufficient cross-party support to secure a second Brexit referendum as fresh doubts were raised over whether Boris Johnson can secure a deal with the EU that can pass through parliament.

The push for a second vote appeared to be gaining momentum before what promises to be a dramatic “super Saturday” showdown in parliament next weekend. That emergency House of Commons sitting, called by Johnson, will be held after a critical EU summit in Brussels on Thursday and Friday. It will coincide with a pro-referendum march through London, which organisers say a million people could join.

On Saturday the chances of Johnson securing a deal that will win the crucial support of the 10 DUP MPs appeared to be in growing doubt as Nigel Dodds, the party’s leader in Westminster, questioned key elements of the proposed deal – including the idea of Northern Ireland being in some form of post-Brexit customs partnership with the EU.

“Northern Ireland must stay in a full UK customs union, full stop,” Dodds said. Asked if the ideas that are reportedly being discussed behind closed doors by the UK government and the EU could work, he added: “No, it cannot work because Northern Ireland has to remain fully part of the UK customs union.”

For Johnson to stand any chance of getting a deal through parliament, he will need the DUP to be firmly behind it. Many Tory MPs in the hardline pro-Brexit European Research Group say they will take their lead from the DUP.

Nigel Dodds and Arlene Foster’s support for any Brexit deal will be crucial to the prime minister. Photograph: Rebecca Black/PA

With the hopes of securing any deal on a knife edge, Labour is understood to be ready to whip its MPs to back a second referendum. “We believe we are getting closer to the majority it needs,” said a source involved in the effort. “The task now is maximising that majority and seeing whether Johnson finally concedes that it is a way out for him too.”

While there is likely to be more support from Tories and ex-Tories for a second referendum if the government fails to secure a deal, and the alternative is a no-deal Brexit, key figures say plans for a referendum on any deal the prime minister returns with is also gaining support.

On Saturday night the Labour MP Peter Kyle said the signs were that a Johnson deal would be bad for the economy and that sufficient MPs would insist on it being subject to a confirmatory public vote. “If Johnson brings back a deal that would deliver less frictionless access to European markets than Theresa May’s deal offered – and that seems to be precisely what is on offer – the idea of that getting through the Commons on its own two feet would be akin to a suicide mission,” he said.

“If a deal like that is offered we will amend it so it can only come into effect after a confirmatory referendum in which the options would be to leave on those terms or Remain. I have no doubt from soundings I have taken that an amendment to secure a confirmatory vote would be successful.”

Pro-referendum MPs are also examining other possibilities including reviving May’s deal and making that subject to a second referendum.

Nick Boles, the former Tory MP who quit over Brexit, said he would back any deal that the EU accepted but made clear that should no deal emerge from the EU council, a second referendum would be needed. “I have been very reluctant to accept that a referendum might be necessary to break the logjam,” he said. “But if Johnson is unable or unwilling to agree a deal with the EU next week, we will be left with no alternative. A snap election will resolve nothing and could prolong the agony.

“Instead we should hold a referendum which offers people the choice between a soft Brexit deal and remaining in the EU. Parliament should pass all the necessary legislation to implement the deal so, if people vote again to Leave, Brexit would happen immediately and require no further votes in parliament.”

Keir Starmer says he will insist on a confirmatory vote. Photograph: Sean Smith/The Guardian

Paul Masterton, a Scottish Tory MP, said: “My instinct on this is the numbers are there for a second referendum if [Johnson] doesn’t bring back a deal, but they aren’t if he does.”

Keir Starmer, the shadow Brexit secretary, told a conference on Saturday: “If Johnson does manage to negotiate a deal… then we will insist that it is put back to the people in a confirmatory vote. If he can’t – or I should say won’t – get a deal … we will take whatever steps are necessary to prevent our country crashing out of the EU without a deal.”

If Johnson has not secured a deal by 19 October, he is obliged under the Benn act to ask the EU for an extension to UK membership until 31 January. Starmer said Labour would ensure he complied with the law. “I have heard some cabinet ministers suggest that Johnson could send a second letter to the EU saying he doesn’t want an extension. That’s the equivalent of attaching a Post-it note to divorce papers saying ‘only kidding’! It’s a ridiculous idea. So, let me be clear: if no deal is secured by this time next week, Boris Johnson must seek and accept an extension.”

Johnson is expected to update the cabinet on Sunday afternoon. A Downing Street source said: “We’ve always wanted a deal. It is good to see progress but we will wait to see if this is a genuine breakthrough. We are a long way from a final deal. The weekend and next week remain critical to leaving with a deal on 31 October. We remain prepared to leave without a deal on 31 October.”