Hunter Biden has become a particular target for Mr. Trump and his allies.

They have focused attention on his personal life, including his 2014 discharge from the Navy Reserve after testing positive for cocaine, and his business career, which included a number of roles that intersected with his father’s political career. He worked with a Delaware-based credit card issuer, at the Commerce Department under President Bill Clinton and as a lobbyist on behalf of various universities, associations and companies.

During his father’s second term as vice president, Hunter Biden increased his international business efforts, including with individuals and entities viewed warily by the United States government and its allies.

In addition to his work in Ukraine for the energy company Burisma, Hunter Biden advised a Romanian businessman with ties to the United States, Gabriel Popoviciu, whose real estate dealings had come under investigation, according to people familiar with the arrangement, which has not been previously reported. The investigation, which came as the United States and its allies were pushing Romania to clamp down on corruption, led to Mr. Popoviciu’s conviction and a prison sentence.

In the deal alluded to by Mr. Trump in the Fox News interview on Sunday, Hunter Biden and a business partner, Devon Archer, were involved in a fund that reportedly pursued an investment from the Chinese government-owned Bank of China.

The fund was announced in late 2013 — days after Hunter Biden and one of his daughters flew to China from Japan aboard Air Force Two with the vice president, who was in the midst of a diplomatic mission intended to calm rising tensions in the region. During the trip, Vice President Biden warned Chinese leaders not to use fighter jets to enforce an air defense zone created by Beijing over contested waters.

The conservative author Peter Schweizer has suggested that Hunter Biden used the trip to secure a deal with the Bank of China — a suggestion that has been echoed by Mr. Trump’s allies in the conservative news media.