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 stated 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 added.

White House spokesperson Kush Desai told the BBC the president exercised his lawful authority to remove Cook, claiming there was cause to remove her due to credible accusations of lying on financial documents.

The Federal Reserve Act does not allow the president to remove a Fed official at will, but permits such action for cause. The allegations against Cook were first made in a public letter from housing finance regulator Bill Pulte, a Trump ally, accusing Cook of falsifying mortgage documents.

No charges have been brought against Cook, and it is unclear if she is under investigation. She previously denied any cause for her dismissal, while legal experts have cast doubt on Trump's justification.

As a key member of the Fed's board, Cook's decisions influence interest rates that affect borrowing and savings in the US. Maintaining her position also serves to uphold the Fed's independence against political pressures.