Mick Mulvaney, the acting White House chief of staff, asked officials in the budget office after President Trump’s July 25 call with the Ukrainian president whether there was a legal justification for withholding hundreds of millions of dollars in military aid to Ukraine, according to two people briefed on an internal White House review.

The discussions, which took place via email in August, came after the hold on the $391 million had already been put in place. Mr. Mulvaney also asked the officials at the Office of Management and Budget how long the aid could be withheld, three people familiar with the review said.

The emails, which were first reported by The Washington Post on Sunday, were surfaced during a review by the White House Counsel’s Office that is examining the events surrounding the Ukraine call. They raise the question of whether Mr. Mulvaney was seeking after the fact to justify the hold, which is central to Democrats’ impeachment investigation into whether Mr. Trump abused his office for political gain, or whether his request was routine.

An administration official on Sunday played down the significance of Mr. Mulvaney’s inquiry, saying that anytime the budget office holds money, it produces a legal justification document. Mr. Mulvaney, the official said, was simply asking to review that document.

The person also hinted at growing tension between the budget office and the White House Counsel’s Office, which the person said had never raised questions about the legality of the hold in real time.

“To be clear,” said Rachel Semmel, a budget office spokeswoman, “there was a legal consensus at every step of the way that the money could be withheld in order to conduct the policy review. O.M.B. works closely with agencies on executing the budget. Routine practices and procedures were followed.”

The White House did not respond to requests for comment.

The overarching purpose of the review by the counsel’s office was to clarify the actions of White House officials involved in the call between Mr. Trump and President Volodymyr Zelensky of Ukraine, and it has helped officials establish a timeline around the withholding of the aid and the call. It originally centered on why John A. Eisenberg, a deputy White House counsel, decided to place a rough transcript of the call in a computer system typically reserved for higher-level secret documents.

On the call, Mr. Trump asked Mr. Zelensky to “do us a favor” and investigate a political rival, former Vice President Joseph R. Biden Jr., and a debunked conspiracy theory that Ukraine had interfered in the 2016 election. The White House has released a rough transcript of the call.

So far, the review has included interviews with a number of White House aides and officials on the National Security Council. The White House Counsel’s Office has also reviewed the system that carries official emails to look for information related to the withheld aid.

Mr. Mulvaney has previously told associates that the aid was released after budget officials raised alarms that the deadline for doing so was approaching.

Officials close to both Mr. Mulvaney and the White House counsel, Pat A. Cipollone — who have been at odds since the beginning of 2019 — expressed frustration that some of the details of the review had leaked.