WASHINGTON — After President Trump told reporters last week that he would like to nominate Herman Cain to fill a spot on the Federal Reserve board, Mr. Cain posted a video to his Facebook page confirming the news. He quickly referred to the sexual harassment allegations that had torpedoed his 2012 presidential campaign, and were already complicating his possible Fed appointment.

“You better believe that the people who hate me, who do not like conservative Republicans, are digging up all the stuff that’s been in storage from eight years ago,” said Mr. Cain, a former restaurant executive who now contributes to a conservative news site and runs a “Super PAC” that supports Mr. Trump.

“So be it! Let them go back and dig up eight-year-old stuff,” he continued. “I will be able to explain it this time, where they wouldn’t let me explain it last time. They were too busy believing the accusers. I’m not mad, y’all. I’m just not going to let the accusers run my life, determine my career.”

Prospective nominees for top government jobs, like the Fed, typically lie low while they wait for Senate confirmation. That has not been the case for Mr. Cain or Stephen Moore, the economic adviser to Mr. Trump whom the president also plans to nominate for a Fed seat.