The hints of irreverence can be traced to a private meeting in Coral Gables, Fla., with the soon-to-be presidential candidate and his rapper acquaintance.

Jeb Bush had grown fond of Pitbull, the Miami performer gone global, who seemed to share his zeal for education policy. But Bush, a former Florida governor, had a question: Why the stage name? The artist replied that a friend had suggested it years ago while they were en route to a pit bull fight.

“Well,” Bush replied at their meeting early this year, “good thing you weren’t on the way to a cockfight.”

As his campaign has struggled to rejuvenate a languishing bid, some close to him have suggested the existence of a (very) secret weapon: It is at least possible that Jeb Bush is funny.

Yet in an election season dominated by the unsubtle zingers and dubious Yiddishisms of Donald Trump — a comedy moment broad enough to accommodate a “Saturday Night Live” hosting gig for Trump and a separate appearance by Hillary Clinton imitating him — Bush’s sensibility has at times proved an odd fit.

He is a candidate so dry that flights of wit can become indistinguishable from a sober default setting.

“Irony doesn’t work,” he said last month aboard his bus in New Hampshire, “in the world of digitized campaigning.”

Playful misfires have included an observation that the star of the new “Supergirl” television show “looked pretty hot” and a quickly abandoned groaner, in one Republican debate, about a town in Iowa with a curious name.

“I was in Washington, Iowa, about three months ago, talking about how bad Washington, D.C., is,” he began, pausing to hear the crickets. “It was — get the — kind of the. Anyway.”

As Bush’s poll numbers have lagged, the bids for laughs have proliferated.

He has tried physical comedy: “Are you doing a photo bomb there?” he asked in Raymond, New Hampshire, striking a hammy pose when someone pointed at a camera. “You’re welcome.”

He has been whimsical with animals: “It’s a girl,” he said Tuesday in Littleton, New Hampshire, surprised to hear that a voter’s large dog was named Amy. “It looks like a boy for some reason.”

He has been wry with schoolchildren curious to hear if he drew inspiration from his famous family members: “Marvin or Neil?” he replied at a charter school in Manchester, New Hampshire, name-checking some nonpresidential Bushes.

And he achieved something approaching self-awareness over the widely mocked title of his “Jeb Can Fix It” bus tour. When a National Review reporter asked on Twitter if Bush could fix her messy room, Bush invited her to email him for help.

“I will empower you to clean up your own room,” Bush wrote back. “It will be liberating! You will feel good about yourself. You will then live a life of purpose and meaning.”

He concluded: “You can do it and I fixed. Jeb.”

For months, Bush’s aides have strained to highlight his humor, ever hopeful that it might lend a bit of verve to his self-serious reputation. (Bush does not make their task easy: He is perhaps the only presidential hopeful to use the phrase “actuarially sound” while trying to charm an Iowa audience.)

His team has posted videos of the candidate musing on the “Sharknado” films and struggling to pull a hooded sweatshirt over his face, labeling the clips “#JebNoFilter.” Staff members cheered a faux striptease at campaign headquarters when Bush dispensed with a smart white button-down to reveal a “Reagan-Bush ‘84” campaign shirt underneath.

And they chuckled along at an Internet meme spawned by Bush standing before a green screen and pointing, allowing editors to place him in assorted fictional scenes: on stage with Kanye West, ruling a disco dance floor, reaching for Michelangelo’s “The Creation of Adam” inside the Sistine Chapel.

“Who says @JebBush doesn’t have a sense of humor!” his top adviser, Sally Bradshaw, asked on Twitter, sharing the article about her boss’ index finger.

The list is long: allies, foes, former aides.

Those who defend his comedic instincts are compelled to hedge, contrasting Bush with his brother George, a wisecracking extrovert.

“Jeb appreciates humor,” Jim Towey, a close friend who served in the administration of George W. Bush, said diplomatically.

“It depends on your definition,” said Mac Stipanovich, a friend and former adviser, when asked if Bush was funny.

“It’s very droll,” Mike Harrell, a longtime golfing partner and Tallahassee lobbyist, said of Bush’s sense of humor. “He looks at you like he just told you his mother is dead.”

Largely spared the rituals of retail politics since his re-election as Florida’s governor in 2002, Bush can have difficulty making his references connect, occasionally losing himself in an analogy.

“What’s the expression?” he asked a New Hampshire audience in September, describing his hard-charging overhauls in Florida. “I’m the bacon in the breakfast experience, not the egg?”

Alas, that was not the expression, at least not one any voters seemed to recognize.

Then there is the down-home flourish — indigenous to North Florida, Bush says — that he has used to discuss his tax plan: “Let the big dog eat.”

So befuddled was an Iowa voter in October that Bush was forced to clarify his position on canine consumption habits.

“I wasn’t talking about big dogs eating other people,” he said at a coffee shop in Oskaloosa. “I had a big dog, and he didn’t eat anybody.”

In certain settings, though, Bush has displayed a disarming irreverence. He has eagerly related the Pitbull conversation to friends and donors. (Pitbull, describing the interaction to Howard Stern in May, rated Bush’s humor as “pretty slick.”)

In September, Bush appeared on “The Late Show With Stephen Colbert,” gamely defending the exclamation point in his campaign logo.

“It connotes excitement,” he said, dryly enough that a question hung over the crowd’s laughs: Was he meekly overexplaining the joke, or firmly in on it?

On “The Tonight Show” in June, Bush likewise played it coy. Agreeing to “slow-jam the news” with Jimmy Fallon, Bush introduced himself and his campaign themes, slipping into Spanish to make his case.

“Hold the teléfono!” Fallon said. “I know you just got back from Miami, but I didn’t think I was interviewing Gov. Pitbull.”

The band set off on a Latin jazz riff, and Fallon shook his hips.

Bush stayed in his chair. The bit was not yet through, and he had agreed to play the straight man.

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