Donald Trump, trailing in opinion polls and facing new accusations by women of unwanted sexual contact, has begun arguing he will win the election on a surge of silent backers who have gone undetected by surveys and the political establishment.

Mr. Trump, who consistently draws thousands of supporters to rallies, said last week his election would be “Brexit all over again,” referring to the unexpected majority of British citizens who voted this year to exit the European Union.

His theory—which others view skeptically—holds that fear of social stigma prevents some voters from admitting they back him, including when talking to pollsters. Last week he called the polls “crooked” and said, “The whole election is being rigged.’’

“This is the only realistic sketch for a Trump victory, and I think there’s some truth to it,” said David Hill, a veteran Republican pollster. He said he saw evidence of Trump voters reluctant to admit their support while working on Jeb Bush’s GOP primary campaign. “But expecting this will be enough to make up 3 points or more?” Mr. Hill said. “That’s a bridge too far.”

Challenges to Mr. Trump’s campaign have mounted as a number of women have accused him of unwanted sexual advances and after a 2005 recording surfaced of him bragging his celebrity allowing him to grope and kiss women. On Friday, a former candidate on “The Apprentice’’ said that Mr. Trump had kissed and touched her without consent in 2007.