President Trump said on Friday he would fire Lisa Cook, a member of the Federal Reserve Board, if she does not resign her post.

Mr. Trump’s threat followed two days after Bill Pulte, the federal housing director who has spearheaded a campaign to oust officials at the Federal Reserve, accused Ms. Cook of engaging in mortgage fraud.

Mr. Pulte, the director of the Federal Housing Finance Agency, alleged that Ms. Cook had falsified records to obtain favorable loan terms and recommended that the Justice Department open a criminal probe. Ms. Cook, in her only comments on the matter, said she had “no intention of being bullied” but pledged to gather “accurate information to answer any legitimate questions and provide the facts.”

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