President Trump is making clear that he fully supports the investigation into Federal Reserve chairman Jerome Powell, saying that the central bank chief may be “crooked.” Mr. Powell is being investigated for renovations being done at the Federal Reserve building which have run over budget. 

Mr. Powell announced on Sunday night that the Federal Reserve had been served with grand jury subpoenas related to his testimony last year about the construction project. He says he did nothing wrong and that he will not bend in the face of any pressure. 

“The threat of criminal charges is a consequence of the Federal Reserve setting interest rates based on our best assessment of what will serve the public, rather than following the preferences of the President,” Mr. Powell said in a video message on Sunday, arguing that monetary policy must not be “directed by political pressure or intimidation.”

On Tuesday, Mr. Trump seemed more than happy to let the investigation into Mr. Powell play out, despite strong reservations from some in his administration and allies on Capitol Hill. 

“Well, he’s billions of dollars over budget, so he either is incompetent or he’s crooked. I don’t know what he is,” Mr. Trump said. “He doesn’t do a very good job.”

Mr. Trump began focusing on Mr. Powell last year after the head of the Federal Housing Finance Agency, Bill Pulte, began posting online about how Mr. Powell was, in his view, being deceptive about the renovations. 

Mr. Pulte would go on to access mortgage records by New York attorney general Letitia James and Federal Reserve board member Lisa Cook. Those accusations from Mr. Pulte served as a pretext for a criminal prosecution of Ms. James and the removal of Ms. Cook from her position. The case against Ms. James has since been dismissed and Ms. Cook remains in her role as her lawsuit plays out. 

Several top Republicans were quick to distance themselves from the new probe into Mr. Powell. Congressman French Hill, who leads the Financial Services Committee which oversees the Fed, said in a statement Monday that the probe is unwarranted. 

“I know Mr. Powell to be a man of integrity with a strong commitment to public service. While over the years we have had our policy disagreements, I found him to be forthright, candid, and a person of the highest integrity,” Mr. Hill, who has known Mr. Powell for more than 30 years, said Monday. “Pursuing criminal charges relating to his testimony on building renovations at a time when the nation’s economy requires focus creates an unnecessary distraction.”

The Senate majority leader, Senator John Thune, also expressed a real sense of concern about this investigation. On Monday, he told reporters that while he did not know the specifics, he hopes this isn’t a pressure campaign against the Fed chairman. 

“I haven’t seen the case or whatever the allegations or charges are, but they better be real and they better be serious,” Mr. Thune said sternly. 

The investigation is already threatening Mr. Trump’s ability to exercise more control over the Federal Reserve, as he has said he wants to do. The president has stated many times that the central bank should take his opinions seriously when making monetary policy changes.

Senator Thom Tillis, a Republican who is retiring, says he will personally hold up the confirmation of new Fed board members if the criminal investigation into Mr. Powell is not “resolved” quickly. As a member of the Senate Banking, Housing, and Urban Affairs Committee, Mr. Tillis has the power to block Mr. Trump’s nominees from coming to the floor if all Democrats vote with him. 

“If there were any remaining doubt whether advisers within the Trump Administration are actively pushing to end the independence of the Federal Reserve, there should now be none. It is now the independence and credibility of the Department of Justice that are in question,” Mr. Tillis said in a statement Sunday night.