"His attempt to fire her, based solely on a referral letter, lacks any factual or legal basis. We will be filing a lawsuit challenging this illegal action," prominent Washington DC lawyer Abbe Lowell said in a statement.
The statement was issued a day after Trump said he would fire Cook for alleged "deceitful and potential criminal conduct" related to mortgages she took out in 2021.
Cook has said Trump does not have the authority to fire her and has vowed to stay on the job.
Trump's showdown with the nominally independent central bank follows other largely successful efforts to bring other elements of the US government under his direct control.
Since returning to office in January, the president has overseen the departure of hundreds of thousands of civil servants, dismantled several agencies and withheld billions of dollars of spending authorised by Congress.
Trump pressured the Fed to lower interest rates during his first term in the White House and he has escalated that campaign in recent months.
The president has demanded that rates be cut by several percentage points and threatened to fire Fed chair Jerome Powell, although he recently backed away from that sabre-rattling.
The attempt to influence the country's central bank has knocked confidence in the US dollar and US sovereign debt and sparked fears of global financial turmoil.
But market reaction to Trump's latest Fed gambit was tame on Tuesday.
Wall Street's main equities indexes were largely flat on the day while the US dollar dropped.
Yields on 2-year, 5-year and 10-year Treasury notes fell, reflecting higher expectations of a near-term rate cut, and rose on longer-dated bonds, in a sign that the Fed's inflation-fighting credentials might weaken.
Trump said in a letter to Cook on Monday that he had "sufficient cause" to fire her because she had described separate properties in Michigan and Georgia as primary residences on mortgage applications before she joined the Fed in 2022.
William Pulte, a Trump appointee who is director of the Federal Housing Finance Agency, first raised questions about Cook's mortgages last week and referred the matter to US Attorney General Pamela Bondi for investigation.
Bondi has yet to say whether the Justice Department will take action.
Trump accused Cook on Monday of having "deceitful and criminal conduct in a financial matter" and said he did not have confidence in her "integrity".
Cook took out the two mortgages in question when she was an academic.
Loans for primary residences can carry lower rates than mortgages on investment properties, which are considered riskier by banks.
Cook listed three mortgages, including two personal residences, on a 2024 financial disclosure form.
She is due to serve on the Fed board through 2038 but the Federal Reserve Act of 1913 allows removal of a sitting governor "for cause".
Until now, that power has not been tested by US presidents, who largely have taken a hands-off approach to Fed matters as a way to ensure confidence in US monetary policy.
