Donald Trump has stepped up his argument that the presidential election will be rigged in favour of Hillary Clinton, claiming that the only way he can lose Pennsylvania is “if cheating goes on”.

Whereas Trump regularly cited opinion polls when he was winning the Republican primaries, his poor showing in recent national surveys has left him sowing doubt about the integrity of the process, even before a vote is cast.

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Speaking at a rally in Altoona, Pennsylvania, on Friday evening, the celebrity tycoon said: “We’re gonna watch Pennsylvania. Go down to certain areas and watch and study and make sure other people don’t come in and vote five times. The only way we can lose, in my opinion – and I really mean this, Pennsylvania – is if cheating goes on. I really believe it.”

Trump went on: “I looked over Pennsylvania and I’m studying it. And we have some great people here. Some great leaders here of the Republican Party, and they’re very concerned about that, and that’s the way we can lose the state, and we have to call up law enforcement, and we have to have the sheriffs and the police chiefs and everybody watching.

“Because if we get cheated out of this election, if we get cheated out of a win in Pennsylvania, which is such a vital state, especially when I know what’s happening here, folks. I know. She can’t beat what’s happening here.”

In 2014, a Pennsylvania judge struck down a law requiring people to produce state-approved photo ID in order to vote. The last Republican presidential candidate to win the state was George H W Bush in 1988, but Trump hopes to defeat Clinton because of its significant blue collar population.

The Republican nominee added: “The only way they can beat it in my opinion – and I mean this 100% – if in certain sections of the state they cheat, OK? So I hope you people can sort of not just vote on the 8th, go around and look and watch other polling places and make sure that it’s 100% fine, because without voter identification – which is shocking, shocking that you don’t have it.”

Trump is trailing by 52% to 42% in Pennsylvania, according to a Quinnipiac survey of likely voters published on Tuesday. Is it one of several battleground states where he has fallen behind since the party conventions and a string of PR disasters.

As his chances of winning the White House appear to recede, Trump has increasingly gone on the offensive against perceived bias in the electoral system in a way no other candidate has in recent times. Earlier this month he told Fox News: “You don’t have to have voter ID to now go in and vote, and it’s a little bit scary, and I’ve heard a lot of bad things. I mean, people are going to walk in, they are going to vote 10 times maybe.”

He told the Washington Post: “If the election is rigged, I would not be surprised.” And last week, campaigning in Ohio, he said he’s “afraid the election is going to be rigged”, without offering evidence.

Several Trump supporters interviewed by the Guardian at a rally in Kissimmee, Florida, on Thursday night expressed the view that defeat, if it happens, will be explicable only in terms of irregularities.

Facebook Twitter Pinterest Donald Trump visits McLanahan Corporation headquarters in Hollidaysburg, Pennsylvania. Photograph: Eric Thayer/Reuters

Studies have shown only a minuscule fraction of voter fraud cases in recent years. Barack Obama has condemned Trump’s claims. “Of course the elections will not be rigged,” he told reporters recently. “What does that mean? The federal government doesn’t run the election process. States and cities and communities all across the country, they are the ones who set up the voting systems and the voting booths.

“And if Mr Trump is suggesting that there is a conspiracy theory that is being propagated across the country, including in places like Texas, where typically it’s not Democrats who are in charge of voting booths, that’s ridiculous. That doesn’t make any sense. And I don’t think anybody would take that seriously.

The president added: “This will be an election like every other election. And I think all of us at some points in our lives have played sports or maybe just played in a schoolyard or a sandbox. And sometimes folks, if they lose, they start complaining that they got cheated. But I’ve never heard of somebody complaining about being cheated before the game was over, or before the score is even tallied. So my suggestion would be go out there and try to win the election.

“If Mr Trump is up 10 or 15 points on Election Day and ends up losing, then maybe he can raise some questions. That doesn’t seem to be the case at the moment.”

During the rally in Altoona, Trump also claimed that the National Security Agency has Clinton’s missing emails and called on the agency to release them publicly. He was joined in Pennsylvania on Friday by Republican national committee chairman Reince Priebus amid speculation of friction between the camps.

Meanwhile Clinton’s husband Bill was campaigning in Las Vegas, Nevada. Addressing a forum of Asian American voters, the former president framed the election as a choice not simply of policy but of America’s identity as a nation of immigrants.

Looking around the room at the thousands who packed an auditorium at the Caesars Palace casino hotel, just down the Las Vegas strip from Trump’s eponymous tower, Clinton said “the metaphor of this election may be walls or bridges.”

“Are we stronger together or stronger apart?” he asked the crowd, comprising mostly of voters representing the nation’s fastest-growing racial group. “You remind people that e pluribus unum -- out of many, one -- is not just a slogan, it’s a way of life.

“You remind people that the eternal effort of making our union more perfect means expanding our definition of us and shrinking our definition of them, not the other way around.”