President Trump accused Democratic presidential front-runner Joseph R. Biden on Monday of enabling his son to profit from foreign connections when he was vice president, while the Biden campaign countered that Mr. Trump is trying to distract from his efforts to pressure Ukraine for dirt on the 2020 rival he most fears.

In the media spotlight of the annual United Nations gathering in New York and through his attorney, Rudolph W. Giuliani, the president blasted Mr. Biden and son Hunter as “corrupt.” They said the younger Mr. Biden raked in millions of dollars from a Ukrainian gas company and more from a venture in China during his father’s eight years in office.

“His son took money from Ukraine, his son took money from China,” Mr. Trump told reporters.

The president declared later in the day, “Joe Biden and his son are corrupt.”

Mr. Giuliani, a former U.S. attorney in New York, took to the airwaves and social media to call on Democrats to investigate the Bidens for “big money selling public office.”

“If Dem party doesn’t call for an investigation of Bidens’ millions from Ukraine and billions from China, they will own it,” Mr. Giuliani tweeted.

The Biden campaign responded that the president is lashing out in desperation against the former vice president. Mr. Biden’s top spokeswoman, Kate Bedingfield, also said Mr. Trump is trying to obscure his efforts to pressure Ukraine’s president for damaging information about the Bidens, allegedly in return for U.S. military aid.

“What we have here is the president of the United States trying to pressure a foreign country to make up stuff about the political opponent at home that he fears the most — Joe Biden,” Ms. Bedingfield said on MSNBC.

In Washington, the Democratic chairs of the House Permanent Select Committee on Intelligence, Oversight and Reform, and Foreign Affairs committees sent a letter Monday to Secretary of State Mike Pompeo demanding documents related to the accusations that the president and Mr. Giuliani pressured the Ukrainian government to investigate the Bidens.

“Seeking to enlist a foreign actor to interfere with an American election undermines our sovereignty, democracy, and the Constitution, which the President is sworn to preserve, protect, and defend,” wrote Democratic Reps. Adam B. Schiff of California, Elijah E. Cummings of Maryland and Eliot L. Engel of New York. “Yet the President and his personal attorney now appear to be openly engaging in precisely this type of abuse of power involving the Ukrainian government ahead of the 2020 election.”

The story has gained in intensity and partisan fury since last week, when a government whistleblower’s complaint came to light over Mr. Trump’s phone call in late July with Ukrainian President Volodymyr Zelensky. The administration has refused to turn over the complaint to congressional Democrats, but an inspector general who reviewed it has deemed the matter “urgent.”

In the call, Mr. Trump reportedly urged the Ukrainian leader to investigate the Bidens. At the time, the U.S. was debating whether to release $250 million in aid for Ukraine, which it later did, and added $140 million.

The president denied Monday that he pressured Mr. Zelensky in exchange for U.S. aid.

“I did not make a statement that ‘You have to do this or I am not going to give you aid,’ ” Mr. Trump told reporters. “I wouldn’t do that. I put no pressure on them whatsoever. I could have.”

Mr. Trump said his conversation with the Ukrainian leader was “a very honorable call.”

“There was no pressure put on them whatsoever,” he said. “It could probably, possibly have been OK if I did, but I didn’t put any pressure on them whatsoever. You know why? Because they want to do the right thing, and they know about corruption. And they probably know that Joe Biden and his son are corrupt. They probably know that. Joe Biden and his son are corrupt.”

Sen. Mitt Romney, Utah Republican and a frequent critic of Mr. Trump, said Monday that such disputes about the conversation’s details indicate that the administration should release the transcript of the call to Congress.

“What has been alleged by some news sources is that the president asked for an investigation into Mr. Biden. That would be in my view very inappropriate and a very serious allegation itself, so let’s find out what exactly what was said,” Mr. Romney said Monday. He added that “it would be very helpful” to see the transcript.

The president and Mr. Giuliani are calling for investigations of Hunter Biden’s pocketing of at least $3 million — $50,000 per month — for a job on the board of the Ukrainian gas company Burisma Holdings. He got the job in 2014 shortly after he was discharged from the Navy Reserves for testing positive for cocaine. His father was playing a prominent role in the Obama administration’s foreign policy.

Hunter Biden, now 49, left the post in April when his term expired.

The $3 million was wired to Hunter Biden through accounts in Cyprus and Latvia, according to reports that The Washington Times did not independently verify.

In 2016, Mr. Biden visited Kyiv as vice president and threatened to withhold $1 billion in U.S. loan guarantees unless the country’s leaders fired Ukraine’s top prosecutor for alleged corruption. The prosecutor also was looking into the gas company, although Hunter Biden apparently was not a target.

The Bidens also have been implicated in a billion-dollar deal in China.

In his book “Secret Empire,” Peter Schweizer described a trip Mr. Biden and his son took to China in 2013 aboard Air Force Two. Less than two weeks later, Hunter Biden’s law firm made a $1 billion deal with a subsidiary of the Chinese government’s Bank of China. The deal later swelled to $1.5 billion, according to the book.

“A lot of money from China,” Mr. Trump said Monday. “China would love nothing they’d rather see than Biden get [elected]. Because they would take this great [trade] deal that we’re about to make and they would really have themselves a deal.”

Mr. Giuliani asked on Twitter how President Barack Obama allowed the Bidens to make “big money selling public office” and whether the Democrats will continue to condone and enable “this kind of pay-for-play.”

Mr. Biden has said he did nothing wrong and that he never discussed the business deals with his son.

Investigative journalist John Solomon unearthed a stream of bank documents that showed more than $3 million flowing from Burisma Holdings to Rosemont Seneca Partners LLC, a hedge fund run by Hunter Biden and his business partner, Devon Archer.

Rosemont received regular Burisma transfers of more than $166,000 a month in 2014 and 2015 while Mr. Biden was Mr. Obama’s point man on Ukraine and as Russian troops were tormenting the country, according to those reports.

The fired prosecutor, Viktor Shokin, told Mr. Solomon that Hunter Biden was on a list of board members who would be interrogated in the Burisma corruption investigation.

In a speech last year at a Washington think tank, Mr. Biden boasted of threatening Ukrainian leaders in 2016 to fire the prosecutor.

“I looked at them and said, ‘I’m leaving in six hours. If the prosecutor is not fired, you’re not getting the money,’ ” Mr. Biden recalled telling then-Ukrainian President Petro Poroshenko.

“Well, son of a bitch, he got fired. And they put in place someone who was solid at the time,” Mr. Biden told the Council on Foreign Relations event, insisting that Mr. Obama was aware of the threat.

Mr. Trump said he is the target of another “Democrat witch hunt” with the whistleblower’s complaint about his phone conversation with the president of Ukraine.

“Here we go again. They failed with Russia, they failed with recession, they failed with everything, and now they’re bringing this up,” Mr. Trump said. “The one who’s got the problem is Biden. Biden did what they would like to have me do, except there’s one problem — I didn’t do it. What Biden did was wrong.”

• Rowan Scarborough and Tom Howell Jr. contributed to this report.

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