Yet in at least two instances, the vice president, Mr. Ayers and other aides have badly overstepped. Mr. Pence recently abandoned an attempt to hire Jon Lerner, a Republican pollster close to Mr. Ayers, as a national security aide, after Mr. Trump discovered Mr. Lerner had helped lead attacks on him in the 2016 election.

The quick dismissal of Mr. Lerner was widely seen as a brushback against Mr. Pence and Mr. Ayers, a way for Mr. Trump’s advisers to signal that they were closely watching the vice president’s office. Two senior White House officials said the Lerner episode made Mr. Trump more acutely aware of what these aides described as Mr. Pence’s empire-building.

Tensions also flared last year, after Mr. Ayers and another Pence aide were reported to have made suggestive comments to Republican donors about planning for an unpredictable 2020 election. Most brazenly, Marty Obst, a senior Pence adviser, told a Republican donor that Mr. Pence wanted to be prepared for the next presidential race in case there was an opening.