Aug 28 (Reuters) - Federal Reserve Governor Lisa Cook filed a lawsuit on Thursday claiming U.S. President Donald Trump has no power to remove her from office, setting up a legal battle that could reset long-established norms for the U.S. central bank’s independence.

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 Republican president has accused Cook of committing mortgage fraud in 2021, a year before she joined the Fed’s governing body.

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 but recently signaled that 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 U.S. 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.