MOSCOW — President Volodymyr Zelensky of Ukraine is already juggling multiple international crises: A war with Russia-backed separatists, an unwanted starring role in the impeachment drama gripping Washington, and tensions with Iran over the downing of Ukrainian jetliner.

Now he is facing growing political turbulence at home.

Prime Minister Oleksiy Honcharuk, an ally of Mr. Zelensky, tendered his resignation on Friday after clandestine audio recordings appeared to show Mr. Honcharuk criticizing the president’s knowledge of economics.

Hours later, Mr. Zelensky rejected Mr. Honcharuk’s resignation in a videotaped meeting that the president’s office posted to Facebook.

The back-and-forth indicated that Mr. Zelensky, a 41-year-old former comedian, faces a power struggle within Ukraine’s elite, despite landslide victories in the presidential election last spring and parliamentary elections last summer.