But there is acknowledgment on both sides of the lectern that some re-examination of the system is warranted, especially at a time when news organizations, which must pay their way to follow the president, are increasingly hamstrung by budget constraints.

“The question really should be, why do you need a protective pool when everybody has cellphones?” said Marlin Fitzwater, who was the press secretary under Presidents Ronald Reagan and George Bush. “When you have a president who can operate a tweet and reach 28 million people from the driveway of any building in America, you don’t really need 14 people sitting there and watching him all night long.” (Mr. Trump’s Twitter account, @realDonaldTrump, has about 18 million followers.)

Mr. Mason said that, since the election, the correspondents’ association and Mr. Trump’s team had “made a lot of progress in forming a protective pool” and that he was confident Mr. Trump would allow reporters to accompany him on Air Force One once he became president.

Mr. Trump’s team has floated the possibility of other changes as well. In his radio interview, Mr. Priebus hinted that the Trump administration might assume control of the seating assignments in the briefing room. The correspondents’ association has decided seating assignments since 1981, in large part because administrations of both parties did not want even the appearance of favoritism in determining press access.

Mr. Priebus’s remarks prompted concern that the new administration would try to usurp some of the association’s control.

Still, many said some kind of seating reform was appropriate.

Mr. McCurry and Ari Fleischer, who served as press secretary under George W. Bush, said they had discussed a setup that would allow a revolving group of journalists into the briefing room rather than reserving seats only for the existing White House press corps. Foreign journalists could attend on Wednesdays, for example, while alternative online media outlets such as Breitbart News and Think Progress could rotate in on Thursdays.

Breitbart — the hard-right website whose former chairman, Stephen K. Bannon, was named Mr. Trump’s chief strategist — is already part of Mr. Trump’s transition pool. The organization’s presence has raised some eyebrows, particularly in liberal media circles, because of its connection to Mr. Bannon. But members of the pools said they did not see it as an issue and pointed out that other partisan news outlets, like the left-leaning Huffington Post, were part of the pool.