“Once she has the job,” Ms. Palmieri added, “her performance, and how hard she works for the people she represents, quells those doubts.”

So Mrs. Clinton will not bluntly ask voters to trust her, aides said. Instead, she will try to own up to the fact that many voters do not, and will discuss this in more personal terms, depending on the setting and audience.

At the same time, aides said, Mrs. Clinton will try to build, or rebuild, trust with voters by demonstrating competence and a devotion to policies that are important to them, like making college more affordable, achieving equal pay for women, and enacting paid family and medical leave.

For Mrs. Clinton, that means emphasizing how she won over skeptical New Yorkers when she ran for the Senate in 2000 after having just moved to the state. “A lot of voters had doubts about me,” she said in the Chicago speech. “I delivered for people, and they learned they could count on me to fight for them. And in the end, I earned their trust.”

Trust is not a problem exclusive to Mrs. Clinton: Never have polls shown the presumptive nominees of both major parties so distrusted or disliked. A Quinnipiac University poll released last week found that only 45 percent of voters considered the presumptive Republican nominee, Donald J. Trump, honest and trustworthy; just 37 percent said so of Mrs. Clinton.

The nearest comparison to her predicament may involve Richard M. Nixon, who polished his image before successfully reintroducing himself to voters in 1968. “He did it by focusing on all the things he brought to the table: foreign policy experience and a different approach to Johnson on Vietnam,” said Kenneth L. Khachigian, a speechwriter for Nixon and later Ronald Reagan.