President Trump said on Monday that he was taking the extraordinary step of removing Lisa Cook from the Board of Governors of the Federal Reserve, in a legally dubious maneuver that could undermine the independence of the nation’s central bank.

Mr. Trump justified the firing, which he said was effective immediately, by pointing to allegations that Ms. Cook may have falsified records to obtain favorable terms on a mortgage. But Ms. Cook and her lawyer vowed to fight her dismissal, maintaining that the president did not have the grounds to oust her.

Ms. Cook has not been charged with wrongdoing or convicted of a crime. In the days before Mr. Trump attempted to remove her, the president had made no secret about his desire to remake the roster of the Fed, as he savaged its members, including Jerome H. Powell, the chair, for keeping interest rates too high.

By targeting Ms. Cook, Mr. Trump appeared to set the stage for a landmark battle that could define the limits of his power over the Fed. Many legal experts raised serious concerns late Monday with the manner of Ms. Cook’s dismissal, and the president’s justification for doing so, as they warned that Mr. Trump’s intervention could compromise an institution at the heart of the economy with damaging results.

In a letter posted to social media, Mr. Trump said the allegations of mortgage fraud undermined Ms. Cook, who was confirmed by the Senate in 2022 as the first Black woman to serve on the Fed’s board. The president claimed she could not perform as an effective financial regulator, as he invoked a power in the Fed’s founding statute that allows him to fire governors for cause.

In a statement released through her lawyer on Monday evening, Ms. Cook said that “no cause exists under the law” for Mr. Trump to fire her.