In the final weeks of Mr. Trump’s term, Mr. Gaetz sought something in return. He privately asked the White House for blanket pre-emptive pardons for himself and unidentified congressional allies for any crimes they may have committed, according to two people told of the discussions.

Around that time, Mr. Gaetz was also publicly calling for broad pardons from Mr. Trump to thwart what he termed the “bloodlust” of their political opponents. But Justice Department investigators had begun questioning Mr. Gaetz’s associates about his conduct, including whether he had a sexual relationship with a 17-year-old that violated sex trafficking laws, in an inquiry that grew out of the case of an indicted associate in Florida.

It was unclear whether Mr. Gaetz or the White House knew at the time about the inquiry, or who else he sought pardons for. Mr. Gaetz did not tell White House aides that he was under investigation for potential sex trafficking violations when he made the request. But top White House lawyers and officials viewed the request for a pre-emptive pardon as a nonstarter that would set a bad precedent, the people said.