While Democrats who control the House are focused on a swift impeachment vote by year’s end, the White House is almost entirely consumed by the trial that would follow in the Republican-controlled Senate, where Mr. Trump’s team believes he would have the chance to defend himself and where Democrats would almost certainly fall short of the two-thirds vote they would need to remove him from office.

That proceeding, however, is also full of unknowns. At a meeting with senior White House officials and senators in the Roosevelt Room of the White House almost three weeks ago, Senator Ted Cruz, Republican of Texas, made clear that there are not enough Senate votes to approve some of the edgier witnesses that Democrats and Republicans want to call. While he mentioned no names, it was interpreted by those in the room to refer to people like Hunter Biden, the son of the former vice president, whom Mr. Trump pushed Ukraine to investigate.