Senator Marco Rubio, the Republican of Florida, has not visited the courthouse in Manhattan to flaunt his support for Donald J. Trump like other potential running mates. He is not a fixture at the former president’s campaign rallies and has not become part of the furniture at Mar-a-Lago, like other Republicans craving relevancy.

Instead, Mr. Rubio has taken a low-key approach in aiming to become the next Republican vice-presidential nominee, a strategy with a clear logic: Mr. Trump is known to bristle when anyone gets too close to his limelight.

But for Mr. Rubio, it’s also a strategy with a history. When the two men competed for the Republican nomination in 2016, Mr. Trump relentlessly mocked his rival’s height, his ears and his mannerisms. Mr. Rubio hurled his own schoolyard taunts, which landed awkwardly and then backfired painfully until his defeat. Since then, the senator has been careful and discreet about how close he gets to Mr. Trump.

His behind-the-scenes maneuvering has transformed him from bitter rival to occasional policy adviser and, now, a leading contender to join Mr. Trump’s ticket, advisers to the former president said.

The son of Cuban immigrants, Mr. Rubio could help Mr. Trump appeal to Hispanic voters. Now more of a seasoned politician than the youthful “Republican savior” on the 2013 cover of Time, Mr. Rubio might also reassure Republican donors and the moderate voters who backed former Gov. Nikki Haley of South Carolina over Mr. Trump in the primary. (Notably, Ms. Haley endorsed Mr. Rubio’s presidential bid in 2016.)

Trump aides and donors also view the senator as one of several candidates who would pose little risk of creating unwanted distractions for a candidate already facing multiple legal threats. He is also known to have a strong relationship with Susie Wiles, a fellow Floridian and Trump campaign senior adviser who is coordinating the search for a running mate.

But it is unclear whether Mr. Rubio’s quiet campaign will work. The soft touch has perplexed Mr. Trump, who has privately wondered how much the senator wants the job, according to two people familiar with the former president’s thinking.

In effect, Mr. Rubio needs to show that he wants the job, without showing that he wants it too much.

Another risk is that anyone in Mr. Trump’s orbit is vulnerable to another round of public humiliation. For Mr. Rubio, the indignities have crept into the conversation.

Mr. Trump has told advisers that Mr. Rubio would have to move out of the state to avoid a potential hurdle: The Constitution potentially bars two residents from the same state from sharing a presidential ticket.

Mr. Rubio has lived near Miami for most of his life and is the father of three college students and a fourth in high school. Mr. Trump moved his residency to Florida in 2019, and has homes in New York and New Jersey. But he has told people that he would not change his address because voters in the state would be too upset to lose him as a resident, according to two people familiar with the conversations.

Mr. Rubio has told people that changing his residency would not be a problem, according to two people familiar with the conversations.

A spokesman for Mr. Rubio declined to comment. A Trump campaign spokesman said that only the former president knows whom he will pick for a running mate.

Mr. Trump told donors at an event this month that Mr. Rubio’s “name is coming up a lot.”

“People love Marco, and I love Marco — he’s a talented guy,” Mr. Trump said.

Kellyanne Conway, a former Trump White House counselor, said Mr. Trump had put Mr. Rubio on his short list in part because of the senator’s reliable support. “He can help buttress the gains Trump is making among key voter groups, be a helpful partner when it comes to governing and be ready to be president on day one,” Ms. Conway said. “He is a prodigious fund-raiser, a foreign policy expert, and facile on TV.”

In the past, Mr. Trump has been known to pit contenders against each other to inform his decision, or, as he has described it, watching allies “fighting over who loves me the most.” While Mr. Rubio has appeared wary to play along, he has started to align himself with Mr. Trump.

Mr. Rubio voted to certify the 2020 elections despite Mr. Trump’s pressure to overturn the results and, at the time, described democracy as “held together by people’s confidence in the election and their willingness to abide by its results.”

Last week, he raised doubts about whether the 2024 contest would be fair and blamed Democrats for undermining the credibility of elections.

“Hopefully, we’ll have a fair election and it will be unquestionable,” he said in a combative appearance on NBC’s “Meet the Press.”

Last month, Mr. Rubio joined several other potential vice-presidential contenders in voting against a $95 billion military aid package for Israel, Taiwan and Ukraine that Mr. Trump also opposed. Despite a record of strongly supporting military allies, Mr. Rubio argued that the money came at the expense of border enforcement and called it “moral extortion.”

“I understand that in our republic, government compromise is necessary — we have to do it all the time,” Mr. Rubio said on the Senate floor, adding that “this is not a compromise. This is legislative blackmail.”

Any discussion of Mr. Rubio’s prospects veers quickly into the brutal 2016 primary, when Mr. Trump — who at 6-foot-3 is about five inches taller than Mr. Rubio and 25 years older — frequently ridiculed the senator as “Little Marco.”

“Looks seem to be very important for Trump, and I have a hard time seeing how he will pick someone who is a foot shorter than him,” said Ana Navarro, a Republican consultant who supported former Gov. Jeb Bush of Florida in 2016 over Mr. Rubio and Mr. Trump. “How will he think that will look good standing next to him and raising their hands together?”

Mocking Mr. Rubio’s height was just one of Mr. Trump’s many insults in the weeks ahead of Florida’s presidential primary in 2016. Mr. Trump also ridiculed Mr. Rubio as a “nervous basket case” with “the biggest ears I’ve ever seen” while hammering away at his Senate attendance record and declaring him weak on illegal immigration but strong on amnesty. Mr. Rubio dished it back, describing Mr. Trump as a small-handed, spray-tanned charlatan who avoided Vietnam because of “injuries from squash.”

The result: Mr. Trump won Florida by nearly 20 percentage points, forcing Mr. Rubio to suspend his presidential campaign and instead seek a second term in the Senate. The two didn’t speak for seven months — until just before the election, when Mr. Trump and his team were nervous about winning the state in the general election, according to three people familiar with conversations.

Mr. Trump’s respect for Mr. Rubio seemed to increase after he had won the White House and Mr. Rubio collected 4.8 million voters in his Senate race, compared with the 4.6 million Floridians who backed Mr. Trump.

For months, whenever they talked, Mr. Trump regularly asked Mr. Rubio how he had won more votes in the state, two of the people said.

It was part of a charm offensive that continued during Mr. Trump’s first weeks in the White House, when he invited Mr. Rubio to dinner at the White House. Mr. Rubio saw an opportunity to influence Mr. Trump on his priorities.

Mr. Rubio continued to advise Mr. Trump on foreign policy, particularly issues related to Venezuela and Cuba, and worked with the Trump administration on expanding the child tax credit and the pandemic aid bill.

This experience could also help reassure traditional Republican voters who have been wary about supporting a second term for Mr. Trump. Mr. Rubio, who turned 53 on Tuesday, has spent roughly half his life in municipal, state and federal elected office.

But while Mr. Rubio’s youth has faded, his ambition and his political upside have not. And that combination has even some Trump critics worried, including former Representative David Jolly of Florida.

“Marco Rubio would win Donald Trump the White House in November,” said Mr. Jolly, who dropped his Republican affiliation because of Mr. Trump’s influence on the party and described himself as “not a fan of Marco.” “He’s the perfect proxy for Haley voters, he speaks to Trumpism without trying to be Trump and he’s been mature and sober. He’s a star, he’s just been a quiet star, lately.”