Donald Trump Jr., the president’s elder son and the member of the family who is most naturally fluent in the language of the Republican base, has frequently been mentioned as a possible candidate for office.

In an interview with The New York Times last fall, as he barnstormed for Republican candidates in the midterm elections, the younger Mr. Trump said of a possible campaign down the road: “Right now, I’m focused on other things, but you never know. I love the intensity of campaigning. I love aspects of the fight. I don’t know how much I would love aspects of the actual job yet.”

A person close to the younger Mr. Trump said he “currently has zero interest in running for office” and that his sole political focus is his father’s campaign next year. Still, the person added that while the younger Mr. Trump does not expect to be a candidate himself, he anticipates being involved in Republican politics “for many years to come.”

Ivanka Trump, the president’s elder daughter, is also seen by West Wing aides and people who have worked with the family as harboring ambitions of her own for elected office. People close to her insist that she has never discussed it, although some of her friends, in private conversations, have been open about the prospect that they see her as a future presidential candidate.

“If she ever wanted to run for president,” Mr. Trump told The Atlantic in a recent interview, “I think she’d be very, very hard to beat.”

Lara Trump, the president’s daughter-in-law, who is actively involved in his re-election campaign, is also mentioned as a possible candidate by some close to the family.

Such comments make clear that the Trump family hopes to extend its influence into the future. And the remarks touched on two lingering questions about the future of the Trumps: Will any of the president’s children run for office, and will he fade quietly if he loses in 2020, or finishes a second term in 2024?