She remains the outright favorite, thanks to an electoral system that is based on state-by-state results. Still, her supporters have reason for concern. According to my colleagues at The Upshot, The New York Times’s data-driven report, Mr. Trump’s chances of victory have jumped to 26 percent as of Monday evening from 10 percent in August. Another respected website, FiveThirtyEight, put his chances at 40.6 percent on Monday morning.

Mrs. Clinton’s Achilles heel is well known: A majority of Americans neither like nor trust her. If Mr. Trump is Teflon, Mrs. Clinton must be Velcro: Every transgression, real or perceived, from her decades-long career in politics stubbornly sticks to her.

When it comes to Mr. Trump, though, his enduring popularity is a mystery.

Mr. Trump was among the first to note his ability to defy the laws of political gravity. “I could stand in the middle of Fifth Avenue and shoot somebody, and I wouldn’t lose voters,” he said in January.

What’s remarkable now is how, after months of withering scrutiny — including accusations of graft, malfeasance and racism — he remains apparently immune to the effects of negative news.

This is a conundrum that has perplexed many New York Times readers who live abroad. “Trump has sleaze written all over his face and behavior,” Neil Douglas, a retired teacher from Canada, wrote in an email that echoed a common sentiment. “Why can’t Americans see through his braggadocio?”

In other countries, and other campaigns, politicians have been derailed by much less. In 2014, the British culture secretary, Maria Miller, resigned over a $10,000 discrepancy in her expense report. In 2012, Mitt Romney’s challenge to Mr. Obama suffered a major blow after he made a comment that criticized 47 percent of the electorate.