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 that instead there would be an internal investigation led by the central bank's inspector general.

President Donald Trump last year said 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 unprecedented 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.