The Chao family’s ties to China have drawn some attention over the years. In 2001, The New Republic examined them in the context of the Republican Party’s softening tone toward the country. When Ms. Chao was nominated as transportation secretary, ProPublica and others highlighted the intersection of her new responsibilities with her family’s business. And in a book published last year, the conservative author Peter Schweizer suggested the Chaos gave Beijing undue influence.

The Times found that the Chaos had an extraordinary proximity to power in China for an American family, marked not only by board memberships in state companies, but also by multiple meetings with the country’s former top leader, including one at his villa. That makes the Chaos stand out on both sides of the Pacific, with sterling political connections going to the pinnacle of power in the world’s two biggest economies.

Ms. Chao’s father, a founder of Foremost in 1964, has for decades cultivated a close relationship with Jiang Zemin, a schoolmate from Shanghai who rose to become China’s president. As the schoolmates crossed paths again in the 1980s, the Chaos reaped dividends from a radar company linked to Mr. Jiang that targeted sales to the Chinese military, documents filed with the Chinese government show.

Though Ms. Chao’s financial disclosure statements indicate she receives no income from Foremost, she made at least four trips to China with the company in the eight years between her job as labor secretary during the George W. Bush administration and her confirmation as transportation secretary in January 2017. And her father accompanied her on at least one trip that she took as labor secretary, in 2008, sitting in on meetings, including with China’s premier, one of the country’s top officials.

Public records show that she has benefited from the company’s success. A gift to Ms. Chao and Mr. McConnell from her father in 2008 helped make Mr. McConnell, the Republican majority leader, one of the richest members of the Senate. And three decades worth of political donations have made the extended family a top contributor to the Republican Party of Kentucky, a wellspring of Mr. McConnell’s power.