A major development in Trump's Fed feud is set to happen next week in the Supreme Court

As the Justice Department continues its investigation into the Federal Reserve, the next front in the central bank's quest to maintain political independence will shift to the Supreme Court.

On Jan. 21, the high court will hear arguments related to President Donald Trump's efforts to fire Federal Reserve Governor Lisa Cook.

Administration officials last year accused Cook of committing mortgage fraud on properties she bought under federally subsidized housing programs. Trump tried to remove Cook from her position but thus far has been prevented by courts. Should the White House win the case, it could set an important precedent for the president's ongoing attempt to oust Fed Chair Jerome Powell.

“If the Court rules against Cook, that would significantly raise the probability that Powell could also be removed based on the DoJ investigation,” Bank of America economist Aditya Bhave said in a note. “We have been arguing that the Cook case is more important for the policy trajectory than the identity of the next Fed Chair. We think that's even more true now.”

The stakes were raised in the Fed saga when Powell announced Sunday that he'd been served with Justice Department subpoenas regarding a controversial multimillion-dollar renovation project at the central bank's headquarters in Washington, D.C. The probe reportedly focuses on whether Powell lied about the project during congressional testimony last year.

Trump has been a fierce Powell critic and toyed last year with the idea of firing him. Powell has insisted that Trump can't remove him, but if the president can establish cause, that could change.

On the surface, the effort to remove Powell seems academic: His term as chair expires in May and Trump then is free to submit a new nominee. However, Powell's term on the Board of Governors runs into 2028, meaning he could stay on and be an obstacle to Trump's efforts to get the Fed to lower rates “MEANINGFULLY!!!,” as he stated in a social media post Tuesday.

However, it also could provide further incentive for Powell to serve out as term on the board as a bulwark against Trump's efforts to thwart the Fed's independence from political manipulation.

For his part, Powell has mostly resisted commenting on Trump's criticism, which has been intensely personal and biting at times. While other presidents have tried to coerce the Fed into easing policy, Trump stands alone in how public and aggressive he has been. Moreover, this is the first time the Justice Department ever has gone after a sitting Fed chair.