Mr. Trump has insisted that his July conversation with Mr. Zelensky was “perfect” even after a reconstructed transcript of the call released by the White House showed him imploring the newly inaugurated Ukrainian leader to “do us a favor” by investigating the Bidens and other Democrats shortly after Mr. Zelensky discussed his need for more American aid to counter Russian aggression.

Undaunted by criticism, Mr. Trump repeated that request on Thursday morning. “I would say that President Zelensky, if it were me, I would recommend that they start an investigation into the Bidens,” Mr. Trump said. “Because nobody has any doubt that they weren’t crooked.”

Even as he seeks investigations by Ukraine and China, the president and Mr. Barr have also solicited help from Australia and Italy to uncover information undermining the origin of Mr. Mueller’s investigation. The Times of London has reported that Mr. Trump sought help from Prime Minister Boris Johnson of Britain, a report that has not been denied by either the White House or the British government.

In Ukraine, Mr. Biden’s son Hunter made $50,000 a month on the board of an energy company. As vice president, his father pressured Ukraine to fire a prosecutor whose office oversaw investigations into the company’s owner. But by the time the elder Mr. Biden acted, there was no public evidence that the prosecutor was actively pursuing an investigation, and no evidence has emerged that the vice president, who was carrying out Obama administration policy, was motivated by his son’s business interests.

In calling for an investigation of American citizens by China, a repressive Communist government with no rule of law, Mr. Trump referred to a business deal Hunter Biden was in that involved a fund drawing investment from the government-owned Bank of China. The fund was announced in late 2013, days after Hunter Biden flew to China aboard Air Force Two with the vice president, who was in the midst of a diplomatic mission.

The president said on Thursday that Hunter Biden was not qualified for that business, noting that he had been discharged from the Navy Reserve after testing positive for cocaine. “He got kicked out of the Navy,” Mr. Trump said. “All of a sudden he’s getting billions of dollars. You know what they call that? They call that a payoff.”

Mr. Trump said he had not asked President Xi Jinping for assistance. “But it’s certainly something we can start thinking about because I’m sure that President Xi does not like being under that kind of scrutiny.”