Federal Reserve governor Lisa Cook has sued President Donald Trump over his attempt to fire her, setting up a potential legal battle with implications for the US central bank’s autonomy.

Cook has asked the court to declare Trump’s firing order “unlawful and void”, and also named Fed Chairman Jerome Powell as defendant.

Trump has said there was “sufficient reason” to believe Cook had made false statements on her mortgage, and cited constitutional powers which he said allowed him to remove her. Cook previously said that “no cause exists under the law” to sack her.

The president has put increasing pressure on the Fed over what he sees as an unwillingness to lower interest rates.

Cook is part of the board responsible for setting interest rates in the US.

Thursday’s lawsuit is likely to bring up a number of legal challenges that could end up at the US Supreme Court.

“This case challenges President Trump’s unprecedented and illegal attempt to remove Governor Cook from her position which, if allowed to occur, would be the first of its kind in the Board’s history,” Abbe Lowell, Cook’s attorney, wrote in the lawsuit.

“It would subvert the Federal Reserve Act … which explicitly requires a showing of ’cause’ for a Governor’s removal, which an unsubstantiated allegation about private mortgage applications submitted by Governor Cook prior to her Senate confirmation is not,” Lowell wrote.

White House spokesperson Kush Desai told the BBC the president “exercised his lawful authority to remove” Cook.

“The President determined there was cause to remove a governor who was credibly accused of lying in financial documents from a highly sensitive position overseeing financial institutions,” he said. “The removal of a governor for cause improves the Federal Reserve Board’s accountability and credibility for both the markets and American people.”

The Federal Reserve Act does not give the president authority to remove a Fed official at will, but as Trump has said, it does allow him to do so “for cause”.

The allegations against Cook were first made in a public letter from housing finance regulator, Bill Pulte, a Trump ally. In the letter he accused Cook of falsifying records to obtain a mortgage.

The letter alleges that she signed two documents, two weeks apart, attesting that two homes in different states were both her primary residence. No charges have been brought against Cook and it is unclear if she is under investigation for these allegations.

Cook’s lawsuit does not address those allegations.

She previously denied that there was any cause to sack her and legal experts have shown scepticism in Trump’s standing.

She is one of seven members of the Fed’s board of governors, and in this position sits on the 12-member committee which is responsible for setting interest rates in the US.

Since returning to Washington, Trump has put increasing pressure on the Fed – especially Powell – over interest rates.

The US president nominates candidates for the role, so removing Cook would mean she could be replaced by someone more favourable to lower interest rates and to the Trump economic agenda.

The Fed’s decision affects the rate at which Americans can borrow money as well as the savings rates on their bank accounts. US interest rates are also closely watched by central banks who set monetary policy in other countries.

Cook voted alongside Powell and most other members of the committee to maintain US interest rates at the Fed’s last rate-setting meeting at the end of July.