“For people like me, who are Republican but reasonable and still have our brains attached, it’s hard to see Trump as a reasonable, sane Republican,” said Dina Vela, a project manager in San Antonio who said she had always voted Republican and remained wary of Hillary Clinton. But to her own surprise, she has started visiting Mrs. Clinton’s campaign website and plans to vote for her.

Since the two parties held their nominating conventions, Mr. Trump’s lead over Mrs. Clinton with Republican women voters has declined by 13 percentage points, according to polls conducted by The New York Times and CBS News.

In late July, 72 percent of Republican women said they would vote for Mr. Trump, a healthy majority, but far below the level won by the past three Republican presidential nominees. In 2012, Mitt Romney won 93 percent of Republican women. In 2008, John McCain won 89 percent, and four years earlier, George W. Bush won 93 percent.

In politically moderate swing states like Pennsylvania, which aides to Mr. Trump say are crucial to his victory, Mr. Trump’s standing with women over all is perilously low among registered voters: Just 27 percent of women back him, compared with 58 percent for Mrs. Clinton, according to a poll by Franklin & Marshall College.