"His attempt to fire her, based solely on a referral letter, lacks any factual or legal basis," Lowell added. "We will be filing a lawsuit challenging this illegal action."
The US Department of Justice (DOJ) last week launched a criminal investigation of alleged mortgage fraud committed by Cook, who is the third political foe accused of the same crime by Trump.
Federal Housing Finance Agency (FHFA) Director William Pulte, a Trump appointee, in April referred New York Attorney General Letitia James to US Attorney General Pam Bondi and her deputy, Todd Blanche, for criminal prosecution.
"His attempt to fire her, based solely on a referral letter, lacks any factual or legal basis."
James, who is also represented by Lowell, successfully sued the president and theTrump Organization for fraud, resulting in a total of $450 million in penalties and interest and a 3-year ban on doing business in the state. Fraudulent activity for which the Trump Organization was found civilly liable included falsifying the valuations of numerous properties. Last week, a state appellate panel upheld the court's findings that the Trump and his organization committed fraud but threw out the 2022 judgment, calling it "excessive."
Earlier this month, the DOJ also issued a subpoena to James as part of a probe into whether she violated Trump's civil rights by filing the fraud suit.
In May, Pulte also referred Sen. Adam Schiff (D-Calif.)—the lead manager of the first of Trump's two impeachments—to the DOJ.
Trump—who has repeatedly railed against "lawfare" and signed a day one executive order aimed at "ending the weaponization of the federal government—has called for the prosecution of James and Schiff.
Responding Monday to the president's allegations and effort to oust her, Cook said that "Trump purported to fire me 'for cause' when no cause exists under the law, and he has no authority to do so."
"I will not resign," she added. "I will continue to carry out my duties to help the American economy as I have been doing since 2022."
Although a president may fire a Fed board member for cause, no such official has ever been removed in the body's 111-year history.
Critics say Trump's effort to terminate Cook is yet another attempt to bully the Fed and Chair Jerome Powell as his administration pushes the nation's central bank to lower interest rates.
However, some observers, including journalist Greg Sargent, warned Tuesday that Trump's firing of Cook could backfire, as her lawsuit's discovery process may allow Lowell "to find out if the White House ordered a Trump loyalist to move against her."
"The real question this raises is: Why is Pulte scrutinizing mortgages that just happen to belong to many high-profile opponents of Trump, and how did he come to select these targets?" Sargent wrote for The New Republic. "Experts recently told me that this use of the FHFA mortgage-fraud process appears highly suspect at best."
Adam Levitin, a professor at the Georgetown University Law Center, recently accused Pulte of "serving as the attack dog for Trump's attempts to gain control over the independent Federal Reserve Board" and targeting a "political enemies list."
"Pulte's actions are something that should scare all of us: If Cook, Schiff, and James can be targeted, what stops Pulte from threatening to review the mortgage application of anyone who speaks out?" Levitin added. "And if politicized mortgage application reviews are somehow okay, won't politicized [Internal Revenue Service] audits be next?"