At the urging of President Trump, Congress is expected to vote this week on whether to cut $500 million per year for public radio and TV stations.

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