This question could become the subject of an intense legal battle.
The Federal Reserve Act does not give the president license to remove a Fed official at will but, as Trump noted in his letter to Ms Cook, it does allow him to do so “for cause”.
Trump cited a “criminal referral” against Cook – after his ally, the housing finance regulator, recently called for Ms Cook to be investigated for mortgage fraud. But it is not clear whether an investigation has been opened, and Ms Cook has not been charged with any crime.
As well as Ms Cook’s denial that there is any cause to sack her, legal experts have shown scepticism. When announcing that a case would be filed over the issue, Cook’s lawyer said the effort by Trump “lacks any factual or legal basis”.
In a statement of its own, the Fed highlighted that there was a high bar for a president to remove a governor for a reason: “Long tenures and removal protections for governors serve as a vital safeguard, ensuring that monetary policy decisions are based on data, economic analysis, and the long-term interests of the American people.”
Market reaction to Trump’s move has been relatively tame so far – as investors appear to be waiting to see if the sacking actually comes to pass. However, there was a sell-off of long-term US government bonds on Tuesday, hinting that some investors were concerned about the Fed’s independence.