The US justice department is dropping its investigation into the Federal Reserve chairman, Jerome Powell, over alleged building cost overruns.

US Attorney Jeanine Pirro said instead there would be an internal investigation led by the central bank's inspector general.

President Donald Trump last year said the cost of the Fed's building renovations had run too high, as part of a long-standing feud with Powell.

Powell's term is nearing its end and the US Senate is currently considering Trump's nominee for his replacement, Kevin Warsh. A key Republican, Thom Tillis, had withheld his support for the nomination unless the Trump administration dropped its investigation.

"American taxpayers deserve answers about the Federal Reserve's fiscal mismanagement, and the Office of the Inspector General's more powerful authorities best position it to get to the bottom of the matter," said White House spokesman Kush Desai in a statement.

"The White House remains as confident as before that the Senate will swiftly confirm Kevin Warsh as the next Federal Reserve Chairman to finally restore competence and confidence in Fed decision-making."

Trump called for the Fed to lower interest rates after returning to office last year, and then began floating the idea of firing Powell - a step some said could be illegal.

Powell took the highly unusual step of releasing a video in January disclosing that the justice department had served the Fed with subpoenas and threatened a criminal indictment over testimony he gave to a Senate committee about renovations to Federal Reserve buildings.

Calling the investigation "unprecedented", Powell said he believed it was opened due to Trump's anger over the Fed's refusal to cut interest rates despite repeated public pressure from the president. Trump said he did not "know anything" about the investigation.

It marked the first time Powell had publicly and robustly pushed back against the US president, as he warned that the independence of the US central bank was at stake.

"This is about whether the Fed will be able to continue to set interest rates based on evidence and economic conditions, or whether instead monetary policy will be directed by political pressure or intimidation," Powell said.