Legally the Fed is an executive branch agency, several of whose governing members are chosen by private banks.

But as an executive branch agency, "The executive power shall be vested in a President of the United States of America" (US Const A2S1) applies, and legally POTUS can do whatever he wants to them.

Practically the courts might (wrongly) rule against it, and who knows how much dirt the banksters have on various officials.