Ms. Warren identifies as a capitalist, though she has called for more far-reaching progressive policies — Medicare for all, for example — than Mr. Biden supports.

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Even before the $15 million figure was released, the Biden campaign and allies were playing down the significance of the quarterly total. One close Biden supporter pointed out the lack of traction that Mr. Sanders and Mr. Buttigieg have had among the broader electorate, as reflected in polling, and said that if money were all that mattered then Democrats would be nominating the billionaire financier Tom Steyer.

Also Thursday, Mr. Biden was asked for his thoughts on selecting a running mate.

“Look, if I could just manufacture someone out of the blue, I would very much think it’s important to have a woman, and even a woman of color, or someone of color as vice president,” he said. “But I’ve learned something different. Not different. Something else. The single most important thing you have to have as a president is someone you know is ideologically compatible with you, that you in fact share the same fundamental concerns and values.”

The Thursday fund-raiser, and the end of the fund-raising quarter, arrived at an uncertain time for Mr. Biden: The House of Representatives is pursuing an impeachment inquiry against Mr. Trump amid revelations that he asked the president of Ukraine to “look into” Mr. Biden and his son Hunter Biden. There is no evidence that as vice president, Mr. Biden intentionally acted to aid his son’s business dealings there, and the news of Mr. Trump’s actions has outraged Democrats and signaled, to some, that the president is concerned about Mr. Biden’s candidacy.

While Mr. Biden is fiercely protective of his family, the moment has thrust the delicate subject into the national spotlight — even as the question of how voters will respond remains fluid, and some of Mr. Biden’s advisers and allies privately acknowledge the risks associated with Mr. Trump’s focus on the Bidens.

That dynamic did not stop his team from fund-raising off the developments involving Mr. Trump. The day after details of the president’s conversation came to light, the campaign tripled its daily online fund-raising average, Mr. Biden’s team said, though an adviser did not respond when asked last week whether that pace had continued.

During the final days of the fund-raising quarter, Mr. Biden spent much of his time at high-dollar fund-raisers, often attending two or even three in one day as he visited wealthy enclaves like Bel Air, Calif., and Park City, Utah. Mr. Biden has been more reliant on high-dollar donors than some of his rivals in the Democratic primary: Ms. Warren and Mr. Sanders have sworn off big-money fund-raisers.