President-elect Donald J. Trump said on Saturday that he wants to replace Christopher A. Wray, the F.B.I. director, with Kash Patel, a hard-line critic of the bureau who has called for shutting down the agency’s Washington headquarters, firing its leadership and bringing the nation’s law enforcement agencies “to heel.”

Mr. Trump’s planned nomination of Mr. Patel has echoes of his failed attempt to place another partisan firebrand, Matt Gaetz, atop the Justice Department as attorney general. It could run into hurdles in the Senate, which will be called on to confirm him, and is sure to send shock waves through the F.B.I., which Mr. Trump and his allies have come to view as part of a “deep state” conspiracy against him.

Mr. Patel has been closely aligned with Mr. Trump’s belief that much of the nation’s law enforcement and national security establishment needs to be purged of bias and held accountable for what they see as unjustified investigations and prosecutions of Mr. Trump and his allies.