Every now and then, while talking to officials in the Oval Office, with friends over dinner, or on the patio at Mar a Lago, President Trump pauses and muses aloud about a subject quietly captivating the Republican Party.

What do you think? JD or Marco?

According to multiple people close to the president, Mr. Trump asks advisers who they prefer, before frequently musing that he should just have Vice President JD Vance and Secretary of State Marco Rubio run together on the same presidential ticket in 2028.

Mr. Trump’s advisers say he is simply having fun polling people, and that 2028 is not at the top of his mind at all. Still, it would be hard for Mr. Trump to ignore that lately, the two men he refers to as “kids” are taking on bigger profiles as the midterm elections approach.

This week, Mr. Rubio appeared in the White House briefing room, affably fielding questions on the Iran war — which his team later produced into a campaign-worthy video. Then he traveled to Italy, popping up alongside Pope Leo XIV and bearing a crystal football as a gift. (“Wow, OK,” Leo, an Augustinian who has taken a vow of poverty, said to Mr. Rubio, a fellow Catholic.) Mr. Rubio also met with Prime Minister Giorgia Meloni of Italy, who, like Leo, has faced diatribes from Mr. Trump over opposition to the war. Next, Mr. Rubio will travel to China with the president.

As Mr. Rubio did his best to charm reporters back in Washington, the White House political operation sent Mr. Vance to a factory in Des Moines to help support a vulnerable House Republican, Representative Zach Nunn. While onstage, Mr. Vance launched into a well-honed attack against Democratic Party policies, and traced his dislike of those policies to his own past shifts in political identity.

“It’s heartbreaking for a kid who came from a union Democrat family to realize that Democrats these days, they seem to care more about gender transition than they do about you keeping more of your hard-earned money,” Mr. Vance told the crowd.

Their dueling appearances, freighted with viral commentary, have generated new speculation about whether Mr. Rubio might eventually challenge Mr. Vance, the presumed front-runner, in a race for the Republican presidential nomination. According to several people close to both men, Mr. Vance and Mr. Rubio, who are friends, do not want to be seen as competing against each other for the 2028 presidential nomination. Others said that it was too early to know how the race would shape up, before seeing how the Republican Party fares in this year’s midterm elections.

“They both appreciate the moment, right?” Senator Eric Schmitt, Republican of Missouri, said in an interview earlier this year about the relationship between Mr. Vance and Mr. Rubio. “This is a historic comeback by a president who was out of office, who came back, and a unique coalition that, you know, brought that to be.”

He added: “I think that drives them to work together in addition to their kind of personal friendship.”

The White House did not comment for this article.

By several accounts, the two actually do get along. It is not uncommon to see them laughing together at White House events. They talk about sports and family when they are together. They are also both very aware of the chatter about their reputations and respective futures.

Mr. Rubio, who serves as national security adviser and acted as chief archivist for nearly a year in addition to his role as the country’s top diplomat, likes to hold up his phone to show friends and colleagues the memes that have been made about him, particularly the ones that comment on the fact that he holds several jobs, according to a person who has seen him do it.

The memes are plentiful, and they have imagined Mr. Rubio in various new roles, depending on the outcome of Mr. Trump’s decisions. He has been cast in the internet’s imagination as Kristi Noem, the former homeland security secretary; the president of Venezuela; and, because Mr. Trump has ordered the Pentagon to release its U.F.O. files, Elliott in the movie “E.T.”

According to his allies, Mr. Rubio’s ubiquity is a sign that he might be able to broaden the MAGA tent beyond Mr. Trump’s red-meat base at a time when the Republican Party is facing serious political headwinds over the economy, the war and aggressive tactics to curb immigration.

“He is a politician who could appeal to a whole lot of Republicans who went along with Trump but weren’t overly enthusiastic about him,” said Whit Ayres, a Republican pollster who worked on Mr. Rubio’s 2010 Senate campaign. “He is very good, not only with English, but with the Spanish language, at framing an argument and making a persuasive case to voters.”

Mr. Rubio has also said that he would stand down if Mr. Vance decided to run: “If JD Vance runs for president, he’s going to be our nominee, and I’ll be one of the first people to support him,” he told Vanity Fair in an interview last year.

Either way, the nomination remains Mr. Vance’s to lose. He is unpopular, with an approval rating sitting at 35 percent in the most recent Washington Post/ABC News/Ipsos poll. But according to a Pew survey taken earlier this year, Mr. Vance is a far more recognizable figure to American voters than most other Trump administration figures — only Mr. Trump and Robert F. Kennedy Jr., the Health and Human Services secretary, are better known.

According to Pew, 75 percent of Republican voters have favorable views of Mr. Vance, compared with 64 percent who have favorable views of Mr. Rubio. And 19 percent of Republican voters have never heard of the secretary of state.

Still, there are signs that some of them are increasingly curious about Mr. Rubio. Sarah Longwell, an anti-Trump Republican strategist, wrote in The Atlantic in April that voters are impressed with Mr. Rubio’s ability to juggle multiple roles at once. In other words, all of those memes might be working.

“The first line of thinking among Rubio’s fans goes something like this: Because he has so many different jobs, he must be competent,” Ms. Longwell wrote, basing her assessment off the weekly voter focus groups she leads. “Another reason voters seem to like Rubio: They see him as the ‘adult in the room.’”

But Mr. Vance, the heir to the president’s political base, is the lone administration figure who can campaign even as he helps Mr. Trump govern. In Iowa, the vice president privately met with influential political leaders, including Jeff Kaufmann, the longtime state party chairman.

“The logistical upper hand that JD has cannot be understated,” Sean Spicer, Mr. Trump’s former press secretary, said on his podcast this week about Mr. Vance’s visit to the state. Mr. Spicer and other allies have said that Mr. Vance’s ongoing work to uncover government fraud and wasteful spending will be popular with voters who have grown increasingly concerned with how their tax dollars are being spent.

Headwinds exist, including the skyrocketing cost, politically and financially, of the Iran war. If the midterm elections do not go well for Republicans, Mr. Vance would be yoked to Mr. Trump’s policies and decisions in a way that any other prospective candidate would not be.

Even though Mr. Vance spoke against Mr. Trump’s decision to attack Iran, he must now defend the president’s decision, which is deeply unpopular with voters and has driven up energy prices. During a visit to a factory in Des Moines, he acknowledged that two Iowans had been killed by an Iranian drone strike on a military base in Kuwait in March.

“We have to make this country worthy of that sacrifice,” Mr. Vance said. As he spoke, he forgot the name of Mr. Nunn’s challenger and asked for assistance “because we’re going to make fun of her a little bit.”

Before traveling to Iowa, Mr. Vance appeared at a fund-raiser for the Republican National Committee. The vice president is the organization’s finance chair, which his allies say puts him in a powerful position to access donors and funding should he decide to run.

But another complicating factor for both men — and, really, for anyone else who decides to run — is Mr. Trump. Never one to shy away from the limelight, Mr. Trump showed an early affinity, during his years on “The Apprentice,” for sitting behind a desk while contestants undermined and undercut each other — all in their quest for the prize.

“The president understandably expects full loyalty from a vice president,” said Marc Short, who was chief of staff to Mr. Vance’s predecessor, Mike Pence. “But in my experience he doesn’t really set up his vice president for political success afterwards.”