Ukrainian officials have for months been threading a needle in discussing the case related to the older Mr. Biden, a leading contender in next year’s presidential election. They have tried to signal to Mr. Trump and his allies that the issues will be investigated, even as they tried to telegraph to Democrats that they were not bending to Mr. Trump’s pressure.

At the same time, the prosecutor’s announcement signaled some attention to an issue Mr. Trump raised in the phone call in July and had repeated publicly: a Ukrainian government investigation of a case touching on a likely opponent in next year’s election.

Mr. Trump’s July phone call with President Volodymyr Zelensky of Ukraine is central to the formal House committee impeachment inquiry called by Speaker Nancy Pelosi. The inquiry is examining whether Mr. Trump betrayed his oath of office and the nation’s security by seeking to enlist the aid of a foreign power to tarnish a political rival.

On the call, Mr. Zelensky suggested that he would assist with an investigation of Burisma, according to White House notes of the call. The Ukrainian president said that a new prosecutor general would soon be appointed who would be “100 percent my person” and would “look into the situation.”

The Ukrainian president said his country was also almost ready to purchase anti-tank missiles, made by Raytheon, to be used to better repel armored assaults by Russian-supported fighters. Mr. Trump responded, “I’d like you to do us a favor, though.”

Mr. Trump has vigorously denied doing anything wrong, and allies of the president said that a reconstructed transcript of the call showed no quid pro quo, making the impeachment inquiry baseless. But Democrats said that Mr. Trump’s request for a favor, and the fact that he had already withheld millions in aid from Ukraine before the call, raised serious questions that must be examined.

On Thursday, the State Department gave initial approval to the $39.2 million sale of 150 Javelin missiles and related equipment to Ukraine. The sale of the javelins to Ukraine must still go through Congress. Ukraine has been fighting Russia for five years in eastern Ukraine since Moscow’s seizure of Crimea.