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.