WASHINGTON — The White House began asking administration officials detailed questions about military assistance to Ukraine after a meeting with President Trump in mid-June, nearly a month before the aid was abruptly frozen, a top Pentagon official told impeachment investigators last month.

Laura K. Cooper, a top Defense Department official in charge of Russia and Ukraine, also testified that she and other Pentagon officials had warned the White House over the summer that continuing to deny Ukraine security assistance that had been approved by Congress could eventually cause the administration to run afoul of the law.

“I had expressed that, you know, because there are only two legally available options and we do not have direction to pursue either,” Ms. Cooper told investigators. “There were many affirmative statements that the Congress has appropriated this, we need to obligate it.”

Ms. Cooper said that she received three questions about the military aid shortly after her office sent out a news release announcing that the Pentagon was planning to provide $250 million to Ukraine for training and equipment. The details of her account emerged on Monday as the House Intelligence Committee released a transcript of her testimony days before it begins holding public impeachment hearings on the Ukraine matter on Capitol Hill.

Ms. Cooper said the White House wanted to know which American industries were involved in the aid, what other countries were contributing to Ukraine, and which government agencies provided the funding.

“We got a question from my chain of command forwarded down from the chief of staff, I believe, from the Department of Defense, asking for follow-up on a meeting with the president,” Ms. Cooper said. “It said ‘Follow-up from POTUS meeting,’ so follow-up from a meeting with the president. So, you know, I’m thinking that the questions were probably questions from the president.”

Ms. Cooper’s testimony is an indication that Mr. Trump was focused on the provision of Ukraine’s security aid at least a month before administration officials were told it was frozen. It also underscored the efforts underway throughout the administration to persuade the president and his staff to release the assistance to Ukraine. Other witnesses have testified that Mr. Trump wanted to use the aid as leverage in order to force Ukraine to announce publicly that it would investigate his political rivals.

Ms. Cooper said that throughout the summer, as Pentagon officials kept sounding the alarm about the legal perils of waiting to provide the aid, the White House, through the Office of Management and Budget, repeatedly asked if they could hold out for longer.

“O.M.B. was trying to see if we could push, you know, keep planning to obligate, but keep pushing the obligations until later in the year and still complete them,” Ms. Cooper testified.

Pentagon officials were concerned, she said, that doing so would violate the Impoundment Control Act, which bars the president from deciding unilaterally not to spend money appropriated by Congress

Ms. Cooper’s closed-door testimony was released as the impeachment inquiry is on the brink of a public phase, with nationally televised public hearings set to begin on Wednesday with appearances by William B. Taylor Jr., the top diplomat in Ukraine, and George P. Kent, a senior State Department official.

The committee also released transcripts of testimony from two diplomats who detailed their growing concerns that Mr. Trump would alter United States policy toward Ukraine because of the influence of Rudolph W. Giuliani, the president’s personal lawyer.

Christopher Anderson, a senior Ukraine specialist at the State Department, recalled hearing warnings from John R. Bolton, the president’s national security adviser, about the influence that Mr. Giuliani was having on Mr. Trump when it came to Ukraine.

“To the best of my recollection, he made a joke about every time Ukraine is mentioned, Giuliani pops up, and that the president was listening to Giuliani about Ukraine,” Mr. Anderson told investigators according to the transcript.

Mr. Anderson, who served as an adviser to Kurt D. Volker, the former United States special envoy to Ukraine, also said that he personally worried that Mr. Giuliani’s presence around the president would make it harder for the United States to force Russia to return to negotiations to end its armed conflict with Ukraine.

“I had the fear that if Giuliani’s narrative took hold, that the Ukrainian government was an enemy of the president, then it would be very hard,” Mr. Anderson said.

Catherine M. Croft, who succeeded Mr. Anderson, said that she was “trepidatious” about taking a job advising Mr. Volker because of Mr. Giuliani. She feared that given his public demands for investigations by Ukraine into former Vice President Joseph R. Biden Jr. and his son, “it was possible that the Trump administration would choose to change its policy to suit domestic politics.”

Mr. Trump railed against the coming hearings on Monday, offering another dose of his combative and unrelenting response to the impeachment inquiry. On Twitter, he accused Representative Adam B. Schiff, Democrat of California and the chairman of the Intelligence Committee, of releasing “doctored transcripts” of the administration officials who testified behind closed doors, though he offered no evidence.

In fact, even the president’s most loyal Republican allies, who participated in hours of closed-door questioning of the witnesses, have refrained from raising doubts about the veracity of the transcripts that have been released so far.

But Mr. Trump said the effort to remove him from office was an “Impeachment Scam,” and he has repeatedly demanded that Republicans join him in undermining the legitimacy of the Democratic-led inquiry. In one tweet on Monday, Mr. Trump called on Republicans to “put out their own transcripts.”

In another, the president called for fraud investigations into Mr. Schiff and the C.I.A. whistle-blower who revealed a July 25 call between Mr. Trump and President Volodymyr Zelensky of Ukraine. Mr. Trump also urged an inquiry into Mark Zaid, the whistle-blower’s lawyer, who has been a critic of Mr. Trump and once tweeted that a “coup has started.”

Mr. Trump has repeatedly said that the July 25 call was “perfect.” On Monday night, Mr. Trump said in a tweet that he planned to release the transcript of an earlier call with Mr. Zelensky, claiming to be the most transparent president in history.

“I am sure you will find it tantalizing!” Mr. Trump wrote.

The transcripts of the three officials released on Monday added more detail to the narrative that Democrats are hoping to present to the public about Mr. Trump’s efforts to use the power of his office to leverage Ukraine in ways that could help him in the 2020 election.

Much of their testimony centered on the decision by the White House to freeze nearly $400 million in military and security aid for Ukraine that had already been approved.

Ms. Cooper testified that the Defense Department had already certified that Ukraine was making “significant forward progress” toward government reform when the White House put an abrupt hold on military aid to that country, citing Mr. Trump’s concerns about corruption.

She testified that the decision left officials across the government baffled about why military assistance crucial to the fight against Russia was being blocked.

“There was a consensus in all of these meetings that this was a government that had a lot of promise, that was tackling corruption, and that we needed to support this government with security assistance,” Ms. Cooper said.

In separate closed-door testimony released on Monday, Ms. Croft told investigators that she received two separate phone calls from alarmed Ukrainian diplomats over the summer quietly inquiring about why the aid had been frozen. The calls came before the president’s hold had been reported publicly, and are further evidence that Ukraine’s government knew what was happening earlier than it said publicly.

“I remember being very surprised at the effectiveness of my Ukrainian counterparts’ diplomatic tradecraft, as in to say they found out very early on, or much earlier than I expected them to,” Ms. Croft said, according to the transcript of her interview.

Ms. Croft also provided investigators new details about another, earlier hold on assistance to Ukraine that was placed by the budget office in the winter of 2017 to 2018. Ms. Croft said that when she was working on Ukraine policy at the White House, Mick Mulvaney, then the director of the budget office and now Mr. Trump’s acting chief of staff, temporarily held up shipments of missiles to Ukraine because of concerns about angering Russia.

Ukraine’s government had for months pressed the Trump administration for the anti-tank missiles, which Ukrainian officials saw as a potential deterrent to further Russian aggression.

Ms. Croft’s testimony offers the first public explanation for why Mr. Mulvaney held up a sale that had already been approved by the State Department and other government agencies. She told investigators that Mr. Mulvaney signed off on the release soon after she and a colleague briefed him on their objectives.

Mark Mazzetti contributed reporting.