Mr. McConnell is not the only member of Ms. Chao’s family raising eyebrows. The House Oversight and Reform Committee is investigating the secretary for possible conflicts of interest involving Foremost Group, a New York-based shipping company owned by her father and sisters. Much of the company’s business is centered in China, and the Chao family, The Times reported in June, enjoys “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.”

The Times noted, “Over the years, Ms. Chao has repeatedly used her connections and celebrity status in China to boost the profile of the company.” In 2017, Ms. Chao’s first official trip to China as a Trump cabinet officer was canceled after ethics concerns about her request to include family members in meetings with Chinese officials were referred to officials in the State and Transportation Departments. The Transportation Department gave no reason for the cancellation at the time, but an official later said the trip conflicted with a scheduled cabinet meeting.

In a Sept. 16 letter, the oversight committee chairman, Elijah Cummings, who died early Thursday, and Raja Krishnamoorthi, head of the subcommittee on economic and consumer policy, reminded Ms. Chao that federal regulations prohibit using one’s public office for “the private gain of friends, relatives, or persons with whom the employee is affiliated in a nongovernmental capacity.”

The committee is also investigating Ms. Chao’s failure last year to divest herself of stock in Vulcan Materials, a producer of construction materials on whose board she served. Before her confirmation in 2017, Ms. Chao signed an ethics agreement to sell her holdings by April 2018. She did not do so until June of this year, just days after The Wall Street Journal reported on the matter. The secretary’s office blames an error by Ms. Chao’s accountant for the lapse.

The committee requested that Ms. Chao provide relevant documents and other information by the end of September. The department has for the most part supplied publicly available material, prompting committee Democrats to consider additional action to compel compliance. Ms. Chao’s office has denied any impropriety and said it will turn over information “on a rolling basis.”

“These false allegations have been hashed over repeatedly in media reports and answered by the department,” said a spokesman. “They are obviously politically motivated.”

Ms. Chao has no official affiliation with or stake in Foremost. But she and Mr. McConnell have benefited handsomely from her family’s good fortunes. In 2008, Ms. Chao’s father gave her and her husband somewhere between $5 and $25 million, according to disclosure reports. Her extended family has contributed more than $1 million to Mr. McConnell’s re-election campaigns.