WASHINGTON — The Transportation Department’s inspector general asked the Justice Department in December to consider a criminal investigation into what it said was Elaine Chao’s misuse of her office as transportation secretary in the Trump administration to help promote her family’s shipping business, which is run by her sister and has extensive business ties with China.

In a report made public on Wednesday, the inspector general said the Justice Department’s criminal and public integrity divisions both declined to take up the matter in the closing weeks of the Trump administration, even after the inspector general found repeated examples of Ms. Chao using her staff and her office to help benefit her family and their business operations and revealed that staff members at the agency had raised ethics concerns.

“A formal investigation into potential misuses of position was warranted,” Mitch Behm, the department’s deputy inspector general, said on Tuesday in a letter to House lawmakers, accompanying a 44-page report detailing the investigation and the findings of wrongdoing.

Ms. Chao, the wife of Senator Mitch McConnell of Kentucky, the Republican leader, announced her resignation on Jan. 7, the day after the Capitol riot. At the time of her departure, an aide to Ms. Chao said her resignation was unrelated to the forthcoming release of the investigation.