As federal prosecutors in Manhattan gathered evidence late last year about President Trump’s role in silencing women with hush payments during the 2016 campaign, Mr. Trump called Matthew G. Whitaker, his newly installed attorney general, with a question. He asked whether Geoffrey S. Berman, the United States attorney for the Southern District of New York and a Trump ally, could be put in charge of the widening investigation, according to several American officials with direct knowledge of the call.