Matthew G. Whitaker, the chief of staff to Attorney General Jeff Sessions, would become the acting No. 2 official at the Justice Department, his White House counterpart, John F. Kelly, told him over the phone on Saturday morning, according to two people briefed on the call. To the White House, he was an obvious choice: a confident former college football player and United States attorney whom Mr. Kelly has privately described as the West Wing’s “eyes and ears” in a department the president has long considered at war with him.