At an annual gathering of tech executives and billionaires in Sun Valley, Idaho, this past July, Gina Raimondo commandeered a table near a duck pond and tried to exert her influence as the U.S. secretary of commerce to help rescue an ailing national champion.

As media moguls and business luminaries cut deals nearby, Ms. Raimondo met with chief executives from Microsoft, Google and other firms and encouraged them to order their semiconductors from the United States, including from Intel. It was important to make more chips in America, she told them, and Intel, the American chip giant, was critical to that effort.

Ms. Raimondo, who is overseeing the Biden administration’s investments in the chip industry, has made similar requests in meetings and phone calls over the past year. She has urged executives at Apple, Amazon, Nvidia, AMD, Marvell Technology and other companies to consider ordering chips from Intel’s U.S. manufacturing plants, according to eight people familiar with those requests, most of whom asked for anonymity to discuss private conversations.

A majority of the firms have rejected her overtures because Intel’s chip-making techniques are not as sophisticated as those of Taiwan Semiconductor Manufacturing Co., the world’s leading chip maker and their primary supplier, these people said.

The doubts expressed by top tech executives show how far Intel has fallen at a moment when Ms. Raimondo is trying to rebuild American chip manufacturing. Her unofficial role as a chip salesperson underscores how much is riding on the success of Intel, a 56-year-old firm that is at the center of President Biden’s efforts to rev up domestic semiconductor production.

While the Biden administration’s endeavor involves many companies, a crucial part of its ambition is pegged to Intel, which lobbied — and helped secure votes — for the 2022 CHIPS Act.