The White House is set to launch its search for the next Federal Reserve chief, according to a senior official, and it will be managed by Gary Cohn, the former Wall Street executive who some market strategists believe could be a candidate for the post himself.

Officials won’t publicly outline any timetable for their decision or shortlist of candidates. Fed Chairwoman Janet Yellen’s term runs through January, and President Donald Trumpdidn’t rule out her reappointment in an April interview.

Ms....