A Democratic power couple featured in hacked emails of the Democratic National Committee will host a fundraiser for Hillary Clinton with President Obama next week on Martha’s Vineyard, Massachusetts.

Hank and Carol Goldberg are holding the fundraiser, which seeks up to $33,400 per couple, at their home in Chilmark, a wealthy enclave where the president and his family are in the midst of a two-week vacation.

In emails released by WikiLeaks last month, the Goldbergs were the subject of an exchange by DNC finance committee staffers as they debated how to schedule a $350,000 fundraiser that the couple wanted to host at their other home, in Chevy Chase, Maryland.

White House officials decided that the drive to suburban Maryland was not a good use of Mr. Obama’s time, and DNC aides proposed to the Goldbergs that they could instead co-host an event with other donors at the Jefferson Hotel, near the White House. The couple decided against that option.

“I think the excitement of hosting at home was a big factor,” wrote a DNC staffer who had to deliver the bad news to the couple.

Instead, on Monday, the Goldbergs will host Mr. Obama at their home on Martha’s Vineyard, the summer playground where the Obamas have vacationed nearly every summer during the president’s two terms. Mr. Goldberg is a real estate executive; Mrs. Goldberg is an artist.

The White House said Mr. Obama will deliver brief campaign remarks to the well-heeled crowd in Chilmark, in which he will no doubt go after Republican nominee Donald Trump, whom he has called “unfit to serve” as president. A small group of pool reporters will be allowed to listen to the president’s brief address but, as usual, will be barred from witnessing Mr. Obama’s question-and-answer session with the donors.

The event will highlight Mr. Obama’s role as the Democratic Party’s biggest draw as Mrs. Clinton seeks to widen her fundraising advantage over Mr. Trump.

It’s not clear whether Mrs. Clinton will attend. She is scheduled to appear at a fundraiser in Edgartown on the island on Aug. 20, the day before Mr. Obama returns to Washington.

The Chilmark event will be Mr. Obama’s 22nd fundraiser this year and his 484th since taking office in 2009, said Mark Knoller of CBS News, the unofficial keeper of presidential statistics.

Mr. Obama, who plans to campaign aggressively for Mrs. Clinton this fall, made his first appearance with the Democratic nominee in North Carolina last month. A few days later, he attended his first fundraiser for Mrs. Clinton in Atlanta, an event to benefit the Hillary Victory Fund, a joint fundraising committee for her campaign and state party organizations.

Mrs. Clinton started August with $44 million cash on hand, about twice as much as Mr. Trump. But the Republican has picked up the pace of fundraising, pulling in $80 million in July, by far his highest total of the race.

The president also gave Mrs. Clinton a full-throated endorsement at the Democratic National Convention in late July. “There has never been a man or a woman — not me, not Bill, nobody — more qualified than Hillary Clinton to serve as president of the United States of America,” he said.

Mr. Obama is a rarity on the campaign trail — a second-term president who is still relatively popular, with a job-approval rating as high as 55 percent.

Mr. Obama also signed a fundraising email this week for Rep. Patrick Murphy of Florida, a Senate candidate in a three-way primary.

“With all that’s at stake, we need Patrick Murphy in the Senate. But he’ll need your help to get there,” Mr. Obama’s email reads. “Patrick stands up to Republicans on behalf of our shared values. It’s why they’re attacking him. They know he can win in November and help Democrats retake the Senate, and they’ll spare no expense to defeat him.”

Mr. Murphy is campaigning for the seat of Republican Sen. Marco Rubio.

The president this summer eclipsed President George W. Bush’s total of 155 fundraisers in his second term but won’t match the more than 400 donor events that Mr. Clinton attended during his second term. He will undoubtedly conduct several more fundraisers this fall in the Democrats’ most reliable cash-generating cities — Los Angeles, with its entertainment industry, and New York, with its Wall Street donors.

In May 2012, during Mr. Obama’s re-election campaign, one fundraiser alone at the home of actor George Clooney pulled in $15 million.

Mr. Trump also is scheduled to appear at two fundraising events in Massachusetts on Saturday — one on Nantucket and another in Osterville.

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