Julie Overgaard, the executive director of South Dakota’s biggest public broadcaster, breathed a sigh of relief in the winter after the state legislature defeated a measure to cut half of her funding.

Then the real battle began.

Ms. Overgaard, 57, is one of dozens of public media executives across the United States bracing for the strong possibility that Congress will vote this week to eliminate funding for public radio and TV stations. The change would have wide-ranging effects on local broadcasters like hers.

“I can’t say that we would be dead immediately, but it would obviously have serious negative impacts,” Ms. Overgaard said.

Congress is expected to vote by the deadline on Friday on a White House proposal, known as a rescission package. It would pull back more than $500 million per year in federal funding that is set to go to the Corporation for Public Broadcasting, the government-backed company that supports public radio and TV stations across the country. Last week, President Trump urged legislators to support the cuts, and threatened to withhold his support for any Republican lawmakers who oppose the proposal.

If the package passes, the federal funding for public media will dry up beginning in October. NPR and PBS would survive — they get a small percentage of their funding from the federal government. But the cuts would force many local stations to sharply reduce their programming and operations. Many public broadcasters receive more than 50 percent of their budgets from the Corporation for Public Broadcasting.

The cuts could even be a death sentence for some stations, which have survived several attempts to choke off funding over the decades.