WASHINGTON — President Trump, already facing impeachment for pressuring Ukraine to investigate his political rivals, publicly called on China on Thursday to examine former Vice President Joseph R. Biden Jr. as well, an extraordinary request for help from a foreign power that could benefit him in next year’s election.

“China should start an investigation into the Bidens, because what happened in China is just about as bad as what happened with Ukraine,” Mr. Trump told reporters as he left the White House to travel to Florida. His request came just moments after he discussed upcoming trade talks with China and said that “if they don’t do what we want, we have tremendous power.”

The president’s call for Chinese intervention means that Mr. Trump and his attorney general have now solicited assistance in discrediting the president’s political opponents from Ukraine, Australia, Italy and, according to one report, Britain. In speaking so publicly on Thursday, a defiant Mr. Trump pushed back against critics who have called such requests an abuse of power, essentially arguing that there was nothing wrong with seeking foreign help to fight corruption.

Throughout his presidency, Mr. Trump has made little effort to hide actions or statements that critics called outrageous violations of norms and standards. And yet because he does them in public, they seem to stir less blowback than if they had been done behind closed doors. Among other things, he repeatedly called on his own Justice Department to investigate his Democratic foes and eventually fired his first attorney general for not protecting him from the Russia investigation.

By boldly repeating the action at the heart of the impeachment inquiry, Mr. Trump almost appeared to dare House Democrats to impeach him. But he left his own party in an increasingly uncomfortable position. Republican lawmakers largely stayed silent on Thursday, neither criticizing the president’s latest comments nor defending them as they nervously awaited other developments that they worried could change the complexion of the case.

Mr. Trump’s comments on China came as the first witness was interviewed by House investigators as part of the impeachment inquiry into the president’s request for investigations into Mr. Biden, a leading contender for the Democratic presidential nomination, and other Democrats during a July 25 phone call with President Volodymyr Zelensky of Ukraine at the same time Mr. Trump was withholding $391 million in American aid.

During a daylong, closed interview, Kurt D. Volker, who resigned last week as the Trump administration’s special envoy to Ukraine, told House staff members about his interactions with the Ukrainians and Rudolph W. Giuliani, the president’s personal lawyer, who has been vigorously lobbying for Ukrainian investigations into Democrats.

Mr. Trump’s comments on Thursday set off a wave of criticism from Democrats, who said he brazenly implicated himself.

“What Donald Trump just said on the South Lawn of the White House was this election’s equivalent of his infamous ‘Russia, if you’re listening’ moment from 2016 — a grotesque choice of lies over truth and self over the country,” Kate Bedingfield, Mr. Biden’s deputy campaign manager, said in a statement.

Ms. Bedingfield was referring to a news conference during the 2016 campaign when Mr. Trump on camera called on Russia to hack Hillary Clinton’s email servers. “Russia, if you’re listening, I hope you’re able to find the 30,000 emails that are missing,” he said. The investigation by the special counsel Robert S. Mueller III later determined that just hours later Russian hackers made their first effort to break into servers used by Mrs. Clinton’s personal office.

Representative Adam B. Schiff of California, the House Intelligence Committee chairman leading the impeachment inquiry, said the president’s latest comments were further evidence of his betrayal of duty.

“The president of the United States encouraging a foreign nation to interfere and help his campaign by investigating a rival is a fundamental breach of the president’s oath of office,” he told reporters.

Mrs. Clinton weighed in as well. “Someone should inform the president that impeachable offenses committed on national television still count,” she wrote on Twitter.

Whether it is impeachable will be decided by the House, which will determine if it amounts to “high crimes and misdemeanors,” as outlined in the Constitution. Such a conclusion does not require a criminal offense, but some critics have argued that the president’s requests for investigations of rivals violate campaign finance laws barring solicitation of foreign contributions.

The Justice Department has not interpreted it that way. In fact, Attorney General William P. Barr himself has been in touch with foreign officials seeking help for an investigation into the origin of Mr. Mueller’s inquiry to determine if it was generated by an illegitimate “hoax” undertaken for partisan reasons, as Mr. Trump has contended.

But the chairwoman of the Federal Election Commission on Thursday seemed to suggest that Mr. Trump’s statements might cross a legal line. Ellen L. Weintraub, a Democrat who has served on the commission since 2002, reposted a statement she made in June warning candidates that “it is illegal for any person to solicit, accept or receive anything of value from a foreign national in connection with a U.S. election.”

Sally Yates, an Obama administration holdover who served as acting attorney general until she was fired by Mr. Trump, said the president behaved as if acting in the light of day would transform corrupt actions into innocent ones. “The president is trying to hypnotize the American people into believing that it can’t be wrong if he says it out loud,” she said.

“The White House counsel in a case like this would find the nearest window and jump out,” said Robert F. Bauer, who served as President Barack Obama’s top lawyer and supports Mr. Biden. “There’s no way to defend it. No way. None.”

Republicans did not even try on Thursday, and some disaffected colleagues said they should speak out against the president. “He’s asking foreign governments to interfere in our election,” former Representative Joe Walsh of Illinois, who is waging a long-shot battle for the Republican nomination, wrote on Twitter. “Right out in the open. Because he thinks he can get away with it. He’s unfit. He’s dangerous.”

Left to defend himself, Mr. Trump weighed back in Thursday night, asserting he was allowed to seek foreign help. “As the President of the United States, I have an absolute right, perhaps even a duty, to investigate, or have investigated, CORRUPTION, and that would include asking, or suggesting, other Countries to help us out!” he wrote on Twitter.

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.”

Mr. Trump’s suggestion came as senior Chinese officials were set to come to Washington next week for another round of trade negotiations. The two countries, locked in a trade war, are hoping to make progress toward a deal after a breakdown in talks in May, leading to an escalation of tariffs.

Mr. Biden is not the only candidate in the race with a child with business in China. Mr. Trump’s elder daughter, Ivanka, a senior White House adviser, has received valuable trademarks from China even after she closed her brand in 2018 because of worsening sales and questions of conflicts of interest.

Nicholas Fandos, Alan Rappeport and Kenneth P. Vogel contributed reporting.

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