Theresa May laid bare the Conservatives’ ambitions to capture some of Labour’s most historic seats in England in a speech on Thursday night, telling voters in Leeds to put aside their traditional allegiances and vote “in the national interest”.

The prime minister told the rally of Conservative campaigners that she needed every vote as a mandate for the difficulties ahead in the Brexit negotiations, citing the German chancellor, Angela Merkel, on the UK’s “illusions” about the process.

“We can see how tough those negotiations are going to be at times,” she said. “We need the strongest possible hand, the strongest possible mandate and the strongest possible leadership as we go into those talks. Yet our opponents are already seeking to disrupt those negotiations – at the same time as 27 other European countries line up to oppose us.”

In a sign of the Conservatives’ bullishness about their chances in Labour’s northern heartlands, May told voters in Harehills that it was the Labour leader, Jeremy Corbyn, on the ballot, not the traditional party. The inner-city suburb was once the seat of Denis Healey, the former Labour chancellor.

“I know this city is one of the places that people call a ‘traditional Labour area’,” May said as she arrived at the rally, wearing the same tartan suit that she wore to announce her Tory leadership bid. “But here – and in every constituency across the country – it may say Labour on the ballot, but it’s Jeremy Corbyn that gets the vote.”

Four Labour MPs have seats in the city, many with sizeable majorities above 7,000. In Leeds East, the constituency that May chose as her venue for the speech, the shadow justice secretary, Richard Burgon, a key Corbyn ally, is defending a 12,533 majority.

Setting out her stall in Harehills, a working-class suburb with high populations of people with Pakistani and east European origins, is the strongest signal yet that May is prepared to be ambitious in her bid to sweep into traditional Labour cities. “This election is not about who you may have voted for in the past, it is about voting in the national interest,” she said.

“Every vote cast for me through my local Conservative candidates in cities like Leeds, and in towns and cities across the UK, will strengthen my hand when I negotiate with the prime ministers, presidents and chancellors of Europe in the months ahead.”

May said next month’s general election would be “the most important election this country has faced in my lifetime – every single vote counts. And everyone in our country has a positive reason to lend me their vote”.

Labour MPs in marginal seats across the country have told the Guardian they have stressed the near-inevitability of the Tories’ victory in June on the doorstep, promising voters that a vote for them will not mean a vote for Corbyn to become the prime minister.

A Jeremy Corbyn supporter in Leeds during the Labour leadership campaign in 2016. The city is one of the party heartlands targeted by May. Photograph: Harry Whitehead/Rex/Shutterstock

People making their way past the shops along the packed terraces off Harehills Lane were incredulous at the prospect of a Conservative victory in their area, even those who said they would consider voting for the party.

There are very few signs that an election is in motion, with no billboards or posters in windows. “I think this election is going to be a whitewash,” said Afraz Khan, his arms crossed by the fruit and vegetables in the International Foods greengrocer. “Brexit is already starting to hit people,” he said, grabbing a handful of his produce.

“These green chillis – we used to get them from Spain, they were £3.49 a kilo. Now they are £4.99 and they come from Pakistan to get the best price. I think it’s a mess. She will win, yes I think so, but not here.”

Aisha Khan, 32, pushing her two-year-old son along Upland Road, said she was astonished that the prime minister had been visiting the area. “Really? What’s she doing in Harehills? Yes, there are places in Leeds the Conservatives could win, I’m sure there are. But not here.”

In Drake’s cycles, a family business on Harehills Lane, 34-year-old Matthew Korlija said he would consider voting for May, though he was undecided whether to vote at all. “I think she probably is best of a bad bunch, she’s better than the other one [Corbyn]. It’s good she’s showing her face to the northerners. But I do think Labour is ingrained here. They won’t win here.”

Mohammed Ali, a 28-year-old solicitor who advises European citizens on their rights post-Brexit, is more of a fan of Jeremy Corbyn. “I think he will do a lot better than people think,” he said. “I think the Muslim community here will remember how she did not condemn the Muslim ban by Donald Trump. I don’t think she will do as well as she says. No way.”

Leeds MPs including former shadow work and pensions secretary Rachel Reeves and former shadow foreign secretary Hilary Benn have both been openly critical of Corbyn in the past. Fabian Hamilton, whose Leeds North East seat encompasses some of the wealthier parts of the city including Roundhay and Alwoodley, is thought to be most at risk of defeat by the Tories.

Can we trust the polls? Most polls wrongly predicted the outcome of the last general election in 2015. This may have been because people changed their mind at the last minute or they were what has been referred to as “shy Tories” – those who felt uncomfortable revealing to pollsters that they intended to vote for the Conservatives. Before the 2015 election, both Mori and ICM polls came up with leads of one point for the Conservatives and one point for Labour. An Ashcroft poll on the day of the election showed a three-point lead for the Conservatives and a YouGov poll on the day suggested a dead heat. The last Populus vote gave Labour a one-point lead. In the end, the Conservatives had a seven-point lead. During the EU referendum campaign, polls did indicate a possible leave vote in the first weeks of June, but edged back to remain in the final days before the vote. Just two of the six polls released the day before the referendum, those carried out by TNS and Opinium, gave leave the edge. This is why Jeremy Corbyn has insisted Labour has a chance of winning despite poor opinion poll ratings.

On Thursday, Labour MP for Exeter Ben Bradshaw wrote an open letter to voters in his local Devon paper, saying: “A vote for me will not affect who governs nationally” in an attempt to put clear water between his record locally and Corbyn’s leadership.

Andrew Gwynne, Labour’s national elections chair, said: “Theresa May is going to extraordinary lengths to blinker the British public and make this election about anything other than her record in government. The people of Leeds won’t be fooled: the only party of working people is the Labour party.

“Under the Tories, working people have picked up the bill, while those at the top have received tens of billions of pounds of tax breaks. Wages have stagnated, public services have suffered huge cuts and our NHS is in crisis.

“It is clearer than ever that the Tories are for the few, not the many. Rather than uniting the country and tackling the challenges we face, their policies are divisive and are taking us backwards.”