WASHINGTON — Herman Cain’s chances to win confirmation to the Federal Reserve plummeted on Thursday as at least four Republican senators indicated they would not back President Trump’s choice to fill a Fed seat.

Those defections would be enough to defeat Mr. Cain in a divided Senate, should Mr. Trump follow through with his plans to formally nominate him.

Mr. Cain, a former pizza magnate whose 2012 presidential run was upended by claims of sexual harassment, is currently undergoing a background check and President Trump has said he will wait for that to be completed before officially nominating him to the seven-member Fed board.

Senator Kevin Cramer, Republican of North Dakota, told reporters on Thursday that he “would vote no” if asked to confirm Mr. Cain. Three other Republican senators — Lisa Murkowski of Alaska, Mitt Romney of Utah and Cory Gardner of Colorado — have also said publicly that they would not support Mr. Cain.