President Trump may be the only member of his family to win major elected office, but his campaign manager for the 2020 re-election effort is already envisioning a family political dynasty that will outlast his time in the White House.

The comments by Brad Parscale, Mr. Trump’s campaign manager, on Saturday were instantly scrutinized for clues as to whether the president’s children, some of whom have been seen by friends and supporters as potential candidates for office, were planning their own political futures.

Mr. Parscale made the remarks at a California Republican convention in Indian Wells, in a speech that touched on the party and the presidency. “The Trumps will be a dynasty that lasts for decades,” Mr. Parscale said.

Asked to explain what that meant, he told reporters in California that he views the Trump family as people with “amazing capabilities.”

“I think you see that from Don Jr. I think you see that from Ivanka. You see it from Jared. You see it from all,” he said, referring to the president’s eldest son, eldest daughter and son-in-law.

Mr. Parscale declined to comment to The New York Times. But a campaign official, who spoke on the condition of anonymity and without official authorization to discuss the speech, said that Mr. Parscale was referring less to possible candidacies in the future and more to activities such as political speeches and fund-raising.

The president and his family have worked aggressively to marginalize or mute whatever anti-Trump sentiment has remained within the national Republican Party. Whether that influence remains once he is out of office — and whether the party reverts to its former shape or adopts a new one — remains an open question.

Longtime supporters of Mr. Trump noted that he and some in his family tended to crave flattery, particularly in public settings, and they chalked up the “dynasty” remark from Mr. Parscale to puffery.

Privately, some Republicans — who did not want to publicly fight with the president’s campaign manager — dismissed Mr. Parscale’s choice of words, noting that the concept of dynastic politics was harmful to Hillary Clinton in 2016 and that it is not a concept historically favored by populists.

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?

Even if no other Trump tries to run a campaign, advisers to Mr. Trump believe he will remain active — on his Twitter feed, in television interviews and potentially at political events of his own.

Few believe he is going to cede the spotlight easily, or follow previous presidents in laying low after leaving office.

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