Jake Gyllenhaal could be any early-30s urbanite. He is sitting in front of me wearing a plain white T-shirt and huge pillowy Nikes, and with a semi-wildman beard and gleaming eyes. He may look normal now, but in his new film, Southpaw, a boxing movie in which he plays a troubled brawler called Billy Hope, he is a huge and sculpted light-heavyweight, a slab of murderous muscle.

The remaking of his own body is becoming a Gyllenhaal signature. His physique in Southpaw is a shocking leap from the skeletal Lou Bloom in Nightcrawler, the sulphurous LA noir about a ghoulish journalist selling footage of freeway pileups to TV news shows. “This transformation,” declared the film’s distributor Harvey Weinstein at Cannes, “is amazing.”

Weinstein was appearing onstage with Gyllenhaal to fire the starting gun for the 2016 Oscars race, telling guests the actor would have his full support in awards season.

Known for his sharp-elbowed campaigning, the mogul announced: “We’ll get revenge” – the wrong to be righted being the Academy’s failure this year to recognise Gyllenhaal’s turn in Nightcrawler. Weinstein declined to comment further for this piece. Gyllenhaal smiles when the subject comes up.

“It’s flattering if someone mentions these things, but frankly it’s also unfortunate because everyone should see a movie and have their own experience. And I know your job is to create context …” He catches himself, keen not to get off on the wrong foot. “Look, it’s Harvey. And Harvey is Harvey. I love him, I really do. But everyone is entitled to their own opinion. And that’s the best I can say about that.”

For Southpaw, there were long months of learning to box, dawn runs, skipping, sparring. The fight scenes still freaked him out. “Even when you’re only filming a fight, even when it’s a movie, in the ring you feel totally naked.” He makes it sound like a sex scene, a monstrous test of nerve. “It’s a strange thing, 700 extras watching.”

Gyllenhaal is likable, inclined to the earnest but self-aware, quick to grin. “The thing was, whenever I took a real hit, my first thought was always: ‘Someone better have got a good fucking angle on that.’ That’s an actor.” He pauses, as he often does. “It may have all been a bit reckless.”

His willingness to get battered may have helped make the part feel like his own. For several years, Southpaw was to have been a vehicle for Eminem; Gyllenhaal was only cast once the rapper dropped out. In that first incarnation, the talk was of a parallel between the star’s hard-knocks youth in Detroit and Billy, a bruiser from a New York orphanage. With Gyllenhaal, the link is less clear: the son of a successful director and screenwriter, he grew up in LA, acting as a child. He appeared in his first film at 11, playing Billy Crystal’s son in City Slickers.

Yet Southpaw is not his first high-profile role as a blue-collar dude in a bad situation. In the Gulf war drama Jarhead, he was a fragile marine; in End of Watch, a sturdy cop. He says the reason the boxing movie endures is because the boxer is such a universal figure, while few of us relate to actors. “Believe me, I take acting seriously, I take it more seriously than …” Again, he reverses his train of thought. “I take it very seriously, let’s just say that. I do also recognise that it is absurd.”

To Dan Gilroy, the writer and director of Nightcrawler, there will have been a reason for the role. “I think Jake was probably attracted to Southpaw as a physical transformation and also as: ‘OK, here’s someone I’ve never played before, from a certain socio-economic background, a place of hardship and struggle.” I’m sure those things were the challenge for him.”

By now, Gilroy says, Gyllenhaal could take his pick from any superhero movie in Hollywood; instead, he makes small, jagged films such as Nightcrawler. The weight loss was the actor’s idea – two stone in 10 weeks. “And to do that he had to fight the industry logic tooth and nail. Because their default position is ‘no’.” Gilroy mimics a nervous executive: “‘What do you mean, you want to lose weight? Don’t lose weight! People know you as Jake, we’re hiring you as Jake! Let me just give you a nice haircut so that you look personable and women will like you.’ That’s the industry. He doesn’t listen. He’s ballsy.”

But there has always been ambition. In 2002, at 21, Gyllenhaal came to London to do theatre, feeling that he should. He starred in the quickfire Kenneth Lonergan play This is Our Youth, at the time the third most famous member of a three-handed cast including Anna Paquin and Hayden Christiansen, then playing Anakin Skywalker in George Lucas’s second batch of Star Wars films.

The play’s director Laurence Boswell was alarmed to discover his only stage experience had been in a school play. He found himself stunned by how good Gyllenhaal was. “My sense was that he was tough enough, ruthless enough, ambitious enough and skilled enough to become a big star.” As well as his talent and his sweetness, Boswell also mentions his “political and strategic skills”, and the industry nous that comes from a Hollywood family. (Boswell mentions his older sister, actress Maggie Gyllenhaal, as a particular influence.)

At 34, having done nothing professionally but act, you wonder if Gyllenhaal feels the job needs to be gruelling to matter. Does he take parts and then find he wants to transform physically – or take parts precisely because they let him stretch and shrink and get weird with his body?

“Does the arrow draw forth the target or vice versa?” He ponders in silence. “Well, I’ve realised I’m fascinated with what it is to be a man. And as a man, physicality is important. I’ve gone on searches …” He sees me squinting uncertainly. “I mean, pushing myself to the limit. With Nightcrawler, I was searching for a physical, chemical state that would come from depriving myself. I was interested in what that would bring out. It wasn’t about losing weight. It was about what happens with a certain kind of deprivation.”

Gyllenhaal has never had a reputation for hedonism, though his career has grazed against tragedy; to many he will always be linked with Heath Ledger, his lover in the pioneering Brokeback Mountain, who would die of a prescription drug overdose in 2008. His own drive sounds like a potent species of workaholism. “He pushes himself harder than I or any director could ever push,” Gilroy says. “He puts more of himself into his characters than any other actor I’ve heard of.”

Should Weinstein win the day, the lesson will surely be that you get an Oscar for playing musclebound orphans called Hope, not ghoulish LA hyper-capitalists. But Gyllenhaal says he thinks Billy might be just as dark as Lou, chewed up by rage. “That was a big reason for me to do this movie – I’m fascinated by my own anger, too.”

He smiles, a vision of Zen tranquility. Really? Does he have a lot of anger? “I would aspire to be someone who is only ever angry at injustice. But frankly, I’m also angry in traffic. And I seriously wonder why. But not just that. There are things that …” He stalls as if there’s something on the tip of his tongue. “I mean, all I can say is: Look, my work expresses best what I’m trying to explore, and you can see from the work I’ve done lately the areas where I’m searching. And the answer to your question is yes, of course, I have anger. Not all the time. But yes. And I’m very curious about it. Not because I go: ‘Ooh, what’s angry like?’ It’s something I experience and want to understand.”I tell him, and it’s true, that as I was sitting with his people waiting to be ushered in here, I Googled his name. Prominent among the results was a MailOnline story detailing the lunch he enjoyed yesterday with a “mystery brunette” in Chelsea. The piece pointed out Gyllenhaal appearing to scratch his back with his fork. “Uncouth,” it sniffed, amid the paparazzi shots taken through the restaurant window. Does any of that make him angry?

He takes a moment. “Well, to be frank, that had yet to sail past my consciousness. But no, I am no longer curious about that stuff. And it doesn’t make me angry. What does make me angry is how easy it is to destroy rather than create …” At this, he embarks on a meandering tangent on the theme of positivity that starts with the flaws of social media, veers into serving on the Cannes jury with the Coen brothers, takes in his enjoyment of the Broadway musical Fun Home, and rounds off with his new film. “Maybe I’m contradicting myself,” he says eventually, “and I hope I am, because the point here was to confuse you.”

A deeply traditional kind of boxing movie, Southpaw comes with a modern twist. Although Weinstein was the public face of its production, much of the finance came from the Chinese entertainment giant Dalian Wanda – owner of, among many other things, cinema chains across Asia and the US. For Gyllenhaal, such deals are part of the eternal balancing act between art and commerce. He was a producer on Nightcrawler, as well as its star; Gilroy thinks he may end up directing, too. His mood right now is, he says, less imperial: “At this point, I’m ecstatic if I’m in a movie that just works, even a little bit. Movies are so hard, just that is miraculous.”