On gun control, after the Las Vegas massacre — in which one Disney employee was shot dead and another was shot in the lung — Mr. Iger said, “In this day and age, we get outraged when an athlete doesn’t stand for the national anthem — where’s the outrage here?”

Then there was the presidential question, which his inquisitor, the Vanity Fair writer Nick Bilton, posed while noting that he had been asked to avoid the subject.

Wincing a bit, the executive mentioned that his wife, the journalist Willow Bay, didn’t love the idea, before saying of his post-Disney future, “I’ll figure it out when I have to figure it out.” (Mr. Iger is scheduled to step down from Disney in 2019.)

It wasn’t a “yes,” but it wasn’t a “no,” either.

I had the chance to catch up with Mr. Iger at a Vanity Fair after-party at the Bouchon Bistro, an outpost for the media and show business elite in Beverly Hills.

I cornered Mr. Iger while he was sitting on a couch drinking a glass of white wine, as waiters delivered trays of lamb sliders and shrimp cocktail to a crowd that included Monica Lewinsky, the HBO chief Richard Plepler and Walter Isaacson, the Aspen Institute president and Steve Jobs biographer.

Brooklyn-born and Long Island-bred, Mr. Iger started out as a production assistant and worked his way up the Disney ladder, with a compensation package of $43.9 million last year. He has a slight New York accent that could help offset his otherwise polished presentation, if he were to run for public office.

I told him I had never seen Disney, or a Disney chief executive, so deep in the political mix. Mr. Iger himself, after all, quit his post on a White House advisory council, in protest of Mr. Trump’s decision to pull out of the Paris climate accords. In September, he called the president’s decision to rescind the Obama-era program allowing the children of undocumented immigrants to remain in the United States — officially called Deferred Action for Childhood Arrivals, or DACA — “cruel and misguided.” (Adding to the sense that Mr. Iger has politics on the brain, records show he switched his party registration from Democrat to no party preference last year.)