The Fed has long maintained independence from the White House to avoid political pressures influencing monetary policy.

Trump’s comments capped a hectic few hours that took his pressure campaign against the Federal Reserve chief to a new level — and sent markets into a shortlived nosedive.

U.S. stocks were shaky after the news broke, as such a move could help Wall Street get the lower interest rates that it loves but would also risk a weakened Fed unable to make the unpopular moves need...