President Trump’s bid to fire a member of the Federal Reserve board is a new escalation of his efforts to amass more power over American government and society: Congress generations ago structured the agency, crucial to the health of the economy, to be independent of White House control.

In purporting to fire the board member, Lisa D. Cook, Mr. Trump is setting up another test of how far the Republican-appointed supermajority on the Supreme Court will let him go in eroding the checks and balances Congress has long imposed on executive power.

His attempt to fire Ms. Cook presents a new twist. It raises the question of whether he alone can decide whether there is cause to fire an official at an independent agency whose leaders are protected by law from arbitrary removal — or whether courts will be willing and able to intervene if judges believe his justification is a pretext.

But the move to oust Ms. Cook, whom the Senate confirmed for a term that ends in 2038, also fits into a now familiar arc, joining the various ways Mr. Trump has systematically accumulated greater authority.

Mr. Trump has stretched the bounds of some legal authorities, like prolifically declaring emergencies to unlock more expansive power, sending troops into the streets of American cities, unilaterally raising import taxes and blocking spending Congress had directed. In this case, he is pushing at the limits of a statute that says Fed board members serve 14-year terms unless removed “for cause” by a president.