President Trump opened a new front in his battle with Lisa D. Cook, the Federal Reserve governor he tried — and failed — to oust from her position last year.

In a letter sent on Wednesday, the White House informed Ms. Cook, who serves on the Board of Governors, that it was “considering” removing her from her position. The letter, which was signed by Dan Scavino, the White House deputy chief of staff, gave Ms. Cook 21 days to respond to previous allegations that she had committed mortgage fraud. Ms. Cook has not been charged with a crime nor convicted of any wrongdoing.

The escalation comes after Mr. Trump suffered a significant legal setback in June when the Supreme Court ruled 5-4 that Ms. Cook, whom Mr. Trump tried to fire last August, should have been given the formal ability to contest the accusations against her. In siding with the lower court, the Supreme Court granted Ms. Cook the ability to continue serving out her term, which is set to end 2038.

But the Supreme Court left much unresolved, leaving open the opportunity for Mr. Trump to revive his attacks. The justices did not clearly articulate the full legal criteria that would allow Mr. Trump to fire Ms. Cook. The Federal Reserve Act stipulates the president can remove a sitting Fed official only “for cause,” which has long been interpreted to mean gross negligence or a dereliction of duty. The justices in June also did not offer an opinion on the exact allegations against her, which revolve around her taking out mortgages on a range of properties that were all listed as a primary residences. Ms. Cook has in the past suggested that these were clerical errors.

“These allegations are as baseless now as they were a year ago when President Trump tried to remove Governor Cook and interfere with the independence of the Federal Reserve,” Abbe Lowell, Ms. Cook’s lawyer, said in a statement. “No matter what President Trump tries to do next, this much is clear under the facts and Supreme Court precedent — there is no valid cause for removing Governor Cook. As we did before, we will challenge this latest pretext and preserve her position and the historic role of the Fed.”

In the letter, which was earlier reported by ABC News, Mr. Scavino said the Fed governor’s actions “may be sufficient to demonstrate that you committed a crime, as you appear to have acquired mortgages that do not meet certain lending requirements and could have received favorable loan terms under fraudulent circumstances.”

“Based on these facts, a jury could find that you intended to defraud these institutions for your own benefit, but at a minimum, this conduct was grossly negligent and demonstrates that you are unfit for the office in which you serve as a controlling member of the Federal Reserve,” Mr. Scavino added in his letter.