WASHINGTON—Senate Republicans warned that attorney general pick Matt Gaetz would struggle to win the support needed for confirmation and said the close ally of President-elect Donald Trump would face intense scrutiny over sexual-misconduct allegations if he continued to seek the post.

Senators and senators-elect barely had time to elect new leadership on Wednesday, giving South Dakota Sen. John Thune the role of majority leader, before that leadership was put to the test with the prospect of confirming controversial former Rep. Matt Gaetz, R-Fla., as the next United States attorney general.