A deep fear came to pass for many artists, museums, and cultural organizations nationwide early Thursday morning when President Trump, in his first federal budget plan, proposed eliminating the National Endowment for the Arts and the National Endowment for the Humanities.

President Trump also proposed scrapping the Corporation for Public Broadcasting, a key revenue source for PBS and National Public Radio stations, as well as the Woodrow Wilson International Center for Scholars.

It was the first time a president has called for ending the endowments. They were created in 1965 when President Lyndon B. Johnson signed legislation declaring that any “advanced civilization” must fully value the arts, the humanities, and cultural activity.

While the combined annual budgets of both endowments — about $300 million — are a tiny fraction of the $1.1 trillion of total annual discretionary spending, grants from these agencies have been deeply valued financial lifelines and highly coveted honors for artists, musicians, writers and scholars for decades.

Nothing will change for the endowments or other agencies immediately. Congress writes the federal budget, not the president, and White House budget plans are largely political documents that telegraph a president’s priorities.

Yet never before have Republicans, who have proposed eliminating the endowments in the past, been so well-positioned to close the agencies, given their control of both houses of Congress and the White House, and now the president’s fiscal plan. Reagan administration officials wanted to slash the endowments at one point, for instance, but they faced a Democratic majority in the House (as well as Reagan friends from Hollywood who favored the endowments).

As for 2017, it is unclear whether Republicans who are friendly to the endowments will fight their own party’s president on their behalf. Mr. Trump went ahead with the proposal even though his daughter Ivanka is a longtime supporter of the arts, and Karen Pence, the wife of Vice President Mike Pence, has been a staunch advocate for art therapy for years, being a painter herself.

Endowment leaders — who, as federal officials, do not lobby the White House or Congress — expressed disappointment that their own executive branch was seeking their demise.

“We are greatly saddened to learn of this proposal for elimination, as N.E.H. has made significant contributions to the public good,” said William D. Adams, chairman of the humanities endowment, in a statement.

Mr. Adams made a point of noting endowment support for preservationist work in Kentucky, the home state of Senate Majority Leader Mitch McConnell, and for theater work by veterans of the wars in Iraq and Afghanistan — a clear overture to Republicans to remember that endowment grants do not only go to liberal elites.

At an impromptu meeting at the arts endowment Wednesday, the chairwoman, Jane Chu, broke the news to the staff and said they would conduct business as usual as the budget-writing process unfolds in Congress, according to three federal officials with knowledge of the meeting, who spoke on the condition of anonymity to disclose the private conversation. Ms. Chu said she called the meeting because she did not want the staff to learn about the proposals from media reports.

Staff members reacted in a professional manner, with a mixture of sadness and some surprise but no visible anger, one of the officials said. Talking points were given to the staff to handle phone calls from endowment grant recipients concerned about their money and the fate of the agencies.

In recent weeks, officials at both endowments were hopeful that President Trump would not propose draconian cuts, let alone elimination. One of the Trump administration’s liaisons to the arts endowment, Mary Anne Carter, had told officials that she was an advocate for the arts and that she wouldn’t have accepted the position if the endowment was going to be eliminated, according to an endowment official who spoke on the condition of anonymity to describe private conversations with Ms. Carter. (She declined to comment on Wednesday.)

Arts groups have already begun a furious lobbying campaign to press Republicans in Congress to save the endowments. The House will draft a budget in the coming months, and arts groups have already been focusing its lobbying efforts there. The Association of Art Museum Directors issued one of the first statements denouncing the president’s plan and urging Congress to save the endowments.

“I’m sort of dumbstruck,” said Brian Ferriso, the association’s president. “I’m hopeful that Congress will take the time to say, ‘Hey, wait a second. We need these cultural elements to our society.’”

Alexandra Nicholis Coon, the executive director of the Massillon Museum in Ohio, said her staff was relying on money from the humanities endowment for a project to record stories, scan letters, and photograph the uniforms of American soldiers who died in World War I. She criticized Mr. Trump’s opposition to the endowments as “shortsighted.”

“It’s disheartening to know that our president and the administration values community memory and the preservation of the American story so little,” said Ms. Coon, whose museum is in Stark County, where Mr. Trump trounced Hillary Clinton in the November election.

While some Republicans have voiced support for the endowments, the arts endowment in particular has been a target of conservatives for decades.

After political battles over the endowments in the late 1980s and 1990s, both agencies created programs and provided grants to more artists and scholars in politically conservative parts of the country, like a rural arts initiative that benefited states like Alabama and North Dakota. Yet endowment money still flowed strongly to liberal-leaning states and cities: New York City arts groups are the largest recipient of federal arts grants.

Some advocates for the arts endowment, which doles out far less money as a percentage than many other governments around the world, have said that its importance is less about the money and more about the message that it sends about the importance of culture in the United States.

PEN America, an advocacy group made up of literary figures, has been circulating a petition in an effort to save the endowments that has already amassed 200,000 signatures, including prominent names such as Salman Rushdie and Margaret Atwood.

Or, as Teresa Eyring of the Theatre Communications Group put it, the time has come for “action mode.”

“This is the beginning of a long road,” said Ms. Eyring, executive director of the group, which represents more than 500 nonprofit theaters around the country. “Now advocates and people in the arts community will communicate with their legislators and really try to make clear the value of this relatively modest but very important investment in our country through the arts.”