A cross-party alliance of opposition politicians has launched an 11th-hour appeal to anti-Tory voters to consider switching allegiance in Thursday’s general election, amid signs that a late surge of tactical voting in a few swing seats could deprive Boris Johnson of a majority in parliament.

The calls from senior Labour, Liberal Democrat and SNP figures come as a major poll suggests Johnson’s likely majority has been cut in half in the last two weeks – from 82 a fortnight ago to just 40 with four days to polling day.

The analysis of almost 30,000 voters, for the pro-EU Best for Britain campaign, also finds that tactical votes by as few as 40,700 people in 36 key seats could prevent Johnson from forming a majority government.

Without a majority, Johnson is unlikely to be able to deliver the central promise of the Tory campaign – “to get Brexit done” – as he will struggle to get enough MPs’ votes. The DUP, which agreed to prop up the Tories after the 2017 general election, is now fiercely opposed to Johnson’s Brexit deal.

The special polling analysis concludes that if tactical voting keeps the Tories out in the three dozen seats, the Conservatives would have 309 MPs, Labour 255, the SNP 49, the Lib Dems 14, Plaid Cymru three and the Greens one. To guarantee a majority, a governing party needs 325 MPs.

Naomi Smith, Best for Britain’s chief executive, said: “This election is on a knife-edge, and, if enough Remainers hold their nose and vote for the candidate with the best chance of stopping the Tories, we’re heading for a hung Parliament and a final-say referendum.”

On Saturday night, there were further signs that Johnson faces a close race against Labour in his Uxbridge and South Ruislip seat, where the Lib Dems are said to be “doing no more campaigning” so as not to take votes from the Labour candidate, Ali Milani.

On Saturday, Vince Cable, the former Lib Dem leader, said he could see the case for tactical voting in some areas. “I have been out there campaigning to persuade Labour voters to support Liberal Democrat candidates in constituencies like Lewes and Cheltenham. But I also know there are a great many good Liberal Democrat voters preparing to back Labour in seats like Canterbury and Sedgefield. I understand why they are doing that and recognise why it may be necessary if we are to stop Brexit being forced on us by the Tories.”

Current leader Jo Swinson, asked in an Observer interview if the Lib Dems should be pulling back in Johnson’s seat, said her job was to drum up votes for the party. But she added: “I understand tactical voting is part of our system, because it’s a rubbish voting system. One way people can use that system to deliver more of what they want is to consider voting tactically.”

Labour’s sitting candidate in Hove, Peter Kyle, a leading advocate of a second Brexit referendum, said: “This is an election like no other, and we either do things differently or pay the price. To deliver for our nation, we need to be smarter. That means encouraging Labour supporters in seats we have no chance of winning – but a non-Tory candidate can win – to do whatever it takes to deny Boris Johnson the majority he craves to deliver his brutal Brexit and continue his reckless premiership.”

Nicola Sturgeon, the SNP leader, said: “Voters can stop a Boris Johnson government if they choose to vote tactically. In Scotland, the SNP is the challenger in every one of the Tory-held seats so by voting SNP, Scotland can play its part in locking Boris Johnson out of Downing Street. In the rest of the UK, it is time to put national interest before tribal party interest so that collectively we can deliver a progressive parliament, focused on protecting our health service, ending austerity, eradicating poverty, delivering justice for women, pensioners and for Scotland, securing the ability to choose our future.”

On Friday, former Tory and Labour prime ministers John Major and Tony Blair attended a rally on tactical voting. Describing the decision to leave the EU as the worst foreign policy choice in his lifetime, Major said: “Tribal loyalty has its place. It can be good and honourable. But sometimes you need to vote with your head and your heart. For your country and your future. This is such a time.”

Saturday’s batch of regular polls showed marked differences in the Tory lead. Opinium’s poll for the Observer has it unchanged from a week ago at 15 points, on 46%, Labour on 31% and the Lib Dems on 13%. But Savanta/ComRes puts Labour up four points on 36%, just six points behind the Tories, who are unchanged on 42%.

Writing in the Observer, Jeremy Corbyn describes the election as a “fight for the survival of our NHS as a free public service that puts patients before profits”. He adds: “My parents’ generation fought hard to establish a universal health service, owned and run by the public. They left it in our trust. It’s our duty to defend it.”

Johnson, in a letter to voters on Sunday, praises the 1945 government for creating the NHS, which he describes as one of the great achievements in British politics in the 20th century. He adds that this election is one of the most important in more than 100 years, the result of which “will be felt for decades to come”.