Would a merger between Mar-a-Lago and Camelot prove irresistible for American voters?

That is a question former President Donald J. Trump has weighed as he considers possible options for his running mate, repeatedly asking advisers and associates in recent weeks about the idea of selecting Robert F. Kennedy Jr. as his No. 2, according to two people familiar with the conversations.

Those close to Mr. Trump do not consider Mr. Kennedy a true contender for the position. Mr. Kennedy, a scion of Democratic royalty, is also already running against Mr. Trump and President Biden as an independent candidate, and he told The New York Times that he would not entertain joining the former president’s ticket.

“I’m flattered by the thought, but it’s not a course I would consider,” Mr. Kennedy said in a text message.

Instead, Mr. Trump’s queries about Mr. Kennedy suggest that the former president remains in the initial stages of his vice-presidential selection process.

Mr. Trump has asked associates about several potential running mates in recent weeks, and while no one knows whom Mr. Trump will choose, he has left some with the impression that he has not yet settled on his first-tier options.

The Trump campaign has begun the early stages of vetting potential contenders, but Mr. Trump does not need to settle on a choice until the Republican National Convention, scheduled to start on July 15 in Milwaukee.

Some people close to Mr. Trump have suggested that choosing a candidate sooner than later might help with fund-raising and campaigning on days he is defending himself in court against the dozens of criminal charges he faces. In 2016, Mr. Trump announced just before the start of his nominating convention in Cleveland that Mike Pence, then the governor of Indiana, would be his running mate.

Mr. Trump’s interest in Mr. Kennedy has centered almost exclusively on the potential power of the branding. He has collected opinions from those around him about whether combining the two famous last names on a single presidential ticket could result in some kind of political magic, according to the people familiar with the conversations.

Mr. Trump has long been intrigued by the Kennedy political dynasty. As president, he often told visitors that he sat at the same Oval Office desk as John F. Kennedy, and he regularly invoked the former president during White House events, including announcements of new tax policies and the ceremonial pardoning of a Thanksgiving turkey.

“I like Trump-Kennedy,” the former president recently told one person. “I like the way that sounds.”

Mr. Trump’s curiosity about Mr. Kennedy as a running mate appeared to undermine the public attacks he and his allies have leveled at the independent candidate.

Last week, Mr. Trump criticized Mr. Kennedy’s views on climate change and the environment, and cast him as more “radical Left” than Mr. Biden. As he did so, he suggested that Mr. Kennedy would siphon more votes from Mr. Biden. “I love that he is running!” Mr. Trump said.

Mr. Kennedy has been running for president as an independent since last year. Last week, he announced Nicole Shanahan, a Silicon Valley lawyer, as his own running mate.

In a Wall Street Journal poll this week across seven battleground states, Mr. Trump had support from 39 percent of voters, compared with 36 percent for Mr. Biden and 11 percent for Mr. Kennedy.

Democrats have signaled more worry about Mr. Kennedy’s potential to play a spoiler role, however, mobilizing a legal and political messaging effort to block him from state ballots and discourage left-leaning voters from supporting him.

While Mr. Kennedy initially tried to challenge Mr. Biden for the Democratic presidential nomination, he has also taken positions more likely to be supported by conservatives, including opposing military assistance for Ukraine in its war against Russia.

Jonathan Swan contributed reporting from Washington, and Rebecca Davis O’Brien from New York.