Every year for the past century, in the green hills of rural Denmark, thousands of Americans and Danes come together to celebrate the Fourth of July.

The gathering is billed as the largest Fourth of July celebration outside the United States, and partygoers decked out in the Stars and Stripes sing American songs, eat hot dogs and, because this is Denmark after all, down Danish beer and aquavit.

This year it’s going to be a little different.

Organizers are expecting the smallest crowd ever. Some locals say they are sitting it out. And the American government has been uninvited, something that has never happened before and in the past would have been unthinkable.

Nixon. Reagan. Walter Cronkite. Walt Disney. Even Dionne Warwick. Some real heavyweights have flocked in on the Fourth to this blip of a town, Rebild, surrounded by purple heather fields and pig farms. Just about every year, the American ambassador to Denmark makes an appearance.

But this year, President Trump’s threats to take over Greenland, a Danish territory, have soured the mood. His obsession with Greenland has hijacked what used to be a very tight Danish-American relationship. Many Danes now question why they should celebrate the most patriotic day in the United States, on Danish soil, with public funds no less.

And so local politicians, no doubt smelling a juicy issue, stepped in.

Lasse Olsen, a municipal council member who has agitated against the event for years, called Mr. Trump “an imperialistic mad man.”

He said that, given the way Mr. Trump had behaved toward Denmark, the presence of any officials from his administration would “disturb and distort” the festival. He led the charge this spring that gave the event’s organizers an ultimatum: Cut American officials from the program or lose logistical support and public funds (typically around $50,000).

The festival organizers agreed to remove U.S. officials from the program and said that embassy officials were sad about it but accepted it. The organizers also said that any Americans coming to the event this weekend in a private capacity were welcome.

The change stings. Organizers plan all year for a gala, a business round table and live music, culminating in an outdoor bash usually with some famous speakers. But this year, all anyone is talking about is Mr. Trump.

“It’s embarrassing,” said Bruce Bro, a retired American businessman with Danish ancestry who is a board member of the Rebild National Park Society, the organization that plans the party. He said that he supported the decision to keep American officials from participating.

“You don’t want people throwing a drink on the ambassador,” he said.

The U.S. Embassy in Copenhagen declined to comment.

The annual pilgrimage to Rebild started in 1912. It was the vision of Max Henius, a Danish American biochemist who helped refine beer brewing. He organized the purchase of land in the hills as a place to celebrate the bonds between Denmark and the United States.

In the early days, Danes who emigrated to America (many to the Midwest, where the landscape reminded them of home) flocked back to Rebild on the Fourth to reconnect with kin. After World War II, when the United States was seen as a hero and ties to the old country were still strong, Mr. Bro said that 50,000 people would show up.

But those ties have faded. And the party in Rebild has taken a hit. Recent crowds rarely surpass the low thousands. This year, organizers say they’ll be lucky to get a thousand.

The program still includes singing the Danish and American anthems, admiring classic American cars, eating hot dogs and open-faced shrimp sandwiches and spotting people dressed as American historical figures.

Last year, Denmark’s foreign minister was the keynote speaker; this time, another minister, lesser known, is planning to come, the Foreign Ministry said.

Lasse Frimand Jensen, mayor of Aalborg, which is one of the municipalities involved in the party, said that his council had decided to take a stand against American officials only after consulting with colleagues in the central government.

The mayor wouldn’t say who, but he’s well connected in the same political party as Mette Frederiksen, the prime minister, who has irritated Mr. Trump by her steadfastness over Greenland. She’s from this same area of Denmark, too.

Over the past few days, organizers have gone through their pre-party rituals: smoothing out American state flags to decorate the party ground, setting up tables and chairs and readying a fleet of golf carts to ferry older people down the gravelly path to the spot in the rugged green valley where the festivities take place.

Organizers say their biggest struggle is to attract younger people.

Mr. Bro said that he heard a lot of people, young and old, saying they didn’t want to come this year because of all the Greenland tensions. As for his own views, he said that he was “horrified” when Mr. Trump threatened to snatch the island from Denmark and that his relatives felt the same way.

“It’s very sad,” he said. “We want to keep this tradition alive. And we blame Trump.”