Cook's lawsuit said Trump, a Republican, violated a federal law allowing the president to remove a Fed governor only for cause when he took the unprecedented step on August 25 of announcing he would fire her.
The president has accused Cook of committing mortgage fraud in 2021, a year before she joined the Fed's governing body.
But Cook's lawsuit said an unsubstantiated allegation about conduct that took place before she was confirmed to her position at the central bank does not amount to cause.
"President Trump's conception of 'cause' has no limiting principle; it would allow him to remove any Federal Reserve Board member with whom he disagrees about policy based on chalked-up allegations," her lawyers wrote in the lawsuit.
The lawsuit also accused Trump of violating her right to due process under the constitution by firing her without notice or a hearing.
The case is likely headed to the Supreme Court, where a conservative majority has at least tentatively allowed Trump to fire officials from other agencies although it recently signalled the Fed may qualify for a rare exception from direct control by the president.
Concerns about the Fed's independence from the White House in setting monetary policy could have a ripple effect throughout the global economy.
The US dollar stumbled against other major currencies after Trump first said he would remove Cook.
A Fed spokesperson said on Tuesday, before the lawsuit was filed, that the Fed would abide by any court decision.
Cook was appointed to the Fed in 2022 by former president Joe Biden.
The law that created the Fed does not define "cause" or lay out any standard or procedures for removal.
No president has ever removed a Fed board member, and the law has never been tested in court.
Several federal laws requiring the president to have cause before removing members of other agencies say that cause can include neglect of duty, malfeasance and inefficiency.
Those laws could be a guide for courts to determine if Trump had cause to fire Cook.
Questions about Cook's mortgages were first raised in August by William Pulte, a Trump appointee who is the director of the Federal Housing Finance Agency.
Pulte referred the matter to Attorney General Pamela Bondi for investigation.
Cook took out the mortgages in Michigan and Georgia in 2021 when she was an academic.
An official financial disclosure form for 2024 lists three mortgages held by Cook, with two listed as personal residences.
Loans for primary residences can carry lower rates than mortgages on investment properties, which are considered riskier by banks.
Some experts have questioned whether transactions that preceded Cook's appointment to the Fed and were in the public record when she was vetted and confirmed by the US Senate could amount to adequate cause to remove her.
The administration could also argue that giving Fed governors any protections from removal violates the president's broad constitutional powers to control the executive branch, as it has in lawsuits filed by other ex-officials.
Trump in an August 25 letter to Cook accused her of having engaged in "deceitful and criminal conduct in a financial matter" and said he did not have confidence in her integrity.
Cook in a statement at the time said that "no causes exist under the law, and (Trump) has no authority" to remove her from the job.
Her lawyers said that Trump's "demands lack any proper process, basis or legal authority."
Cook's departure would allow Trump to name his fourth pick to the Fed's seven-member board.
