Questions about the independence of the program’s inspector general emerged almost immediately after the president signed the $2 trillion economic relief package into law.

In an unusual signing statement, Mr. Trump suggested he had the power to decide what information a newly created inspector general intended to monitor the fund could share with Congress. That prompted concern among lawmakers and watchdog groups, which said Mr. Trump’s statement went further than previous presidents in limiting the authority of the inspector general.

The president has not been shy about his resistance to independent oversight in recent weeks, particularly individuals who played a role in the impeachment proceedings.

On Friday night, after announcing that he intended to nominate Mr. Miller, Mr. Trump also said that he was firing the intelligence community’s inspector general who alerted lawmakers to a whistle-blower complaint about the president’s dealings with Ukraine.

At a news conference on Monday, Mr. Trump assailed a Department of Health and Human Services inspector general for writing a report detailing equipment shortages that hospitals face as they confront the coronavirus. Mr. Trump suggested that the report was politically motivated because the inspector general was first appointed during the Obama administration.

Mr. Trump further eroded the oversight architecture of the $2 trillion economic stabilization package on Tuesday when he removed Glenn A. Fine, the Defense Department’s acting inspector general who had been tapped to lead a separate Pandemic Response Accountability Committee, and replaced him with Sean O’Donnell, who is the inspector general of the Environmental Protection Agency. Mr. Fine had been selected to lead the committee by a panel of inspectors general.

A White House spokesman declined to make Mr. Miller available for an interview and would not comment on whether he had a role in drafting the signing statement or discuss his level of involvement in the impeachment process.