Though he has yet to call for impeachment proceedings against Mr. Trump — as have several of his rivals for the 2020 Democratic nomination — Mr. Biden on Saturday tiptoed closer to embracing the idea, which has been gaining support on Capitol Hill despite opposition from House Speaker Nancy Pelosi.

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Mr. Biden, whose appearances on the campaign trail can be halting and sprinkled with misstatements, has generally delivered his strongest performances when focused on Mr. Trump. Speaking about the president allows Mr. Biden to discuss foreign policy and national security, issues that his campaign has said differentiate Mr. Biden, a former Senate Foreign Relations Committee chairman, from the rest of the 2020 Democratic field.

While Mr. Trump’s attacks give Mr. Biden the one-on-one showdown with the president that his campaign has spent months trying to create, it also exposes him and his son to another round of questions about Hunter Biden’s business activities in Ukraine.

The Biden campaign moved quickly to warn the news media over the story, underscoring a deep concern about how allegations about the younger Mr. Biden’s work will be received by voters. “Any article, segment analysis and commentary that does not demonstrably state at the outset that there is no factual basis for Trump’s claim, and in fact that they are wholly discredited, is misleading readers and viewers ,” said the deputy campaign manager, Kate Bedingfield, in an email to reporters.

But Biden advisers also seized on the furor to portray Mr. Trump as fixated on, and worried about, a potential general election race against Mr. Biden.

“There is only one candidate the president is trying to get foreign governments to dig up bogus dirt on,” Anita Dunn, a senior adviser to Mr. Biden, said.

The effort by the president and his team to shift the focus to Mr. Biden could boomerang, casting the Democratic front-runner as a sympathetic figure unfairly attacked with foreign help. It could just as easily mark a defining moment for Mr. Biden, a 76-year-old politician first elected to the Senate in 1972 and long accustomed to playing by the more genteel political rules of a different era.