A conservative firebrand on Capitol Hill, Mr. Gaetz could take over a Justice Department that Mr. Trump has already pledged to drastically overhaul.

The department with roughly 115,000 employees sits at the center of Mr. Trump’s key campaign promises, including reducing crime and dolling out billions of dollars in grant funding for immigration enforcement, border security and policing.

Mr. Gaetz last year orchestrated the ouster of then-House Speaker Kevin McCarthy, California Republican, after he struck a deal with Democrats to avoid a government shutdown. He has also been a stalwart loyalist of Mr. Trump since he first came to Washington.