“For some, it is for ideological reasons, and for others it is for stylistic reasons,” Mr. Dent said, complaining of the “exhausting” amount of “instability, chaos and dysfunction” surrounding Mr. Trump.

Mr. Pence has made no overt efforts to separate himself from the beleaguered president. He has kept up his relentless public praise and even in private is careful to bow to the president.

Mr. Pence’s aides, however, have been less restrained in private, according to two people briefed on the conversations. In a June meeting with Al Hubbard, an Indiana Republican who was a top economic official in Mr. Bush’s White House, an aide to the vice president, Marty Obst, said that they wanted to be prepared to run in case there was an opening in 2020 and that Mr. Pence would need Mr. Hubbard’s help, according to a Republican briefed on the meeting. Reached on the phone, Mr. Hubbard declined to comment.

Mr. Ayers has signaled to multiple major Republican donors that Mr. Pence wants to be ready.

Mr. Obst denied that he and Mr. Ayers had made any private insinuations and called suggestions that the vice president was positioning himself for 2020 “beyond ridiculous.”

For his part, Mr. Pence is methodically establishing his own identity and bestowing personal touches on people who could pay dividends in the future. He not only spoke in June at one of the most important yearly events for Iowa Republicans, Senator Joni Ernst’s pig roast, but he also held a separate, more intimate gathering for donors afterward.

When he arrived in Des Moines on Air Force Two, Mr. Pence was greeted by an Iowan who had complained about his experience with the Affordable Care Act — and who happened to be a member of the state Republican central committee.

The vice president has also turned his residence at the Naval Observatory into a hub for relationship building. In June, he opened the mansion to social conservative activists like Tony Perkins of the Family Research Council and representatives of the billionaire kingmakers Charles G. and David H. Koch.