Aaron Eckhart's day began as they all do: an hour in the gym, then same again with a cigar. Jetlag cannot disrupt that, snow can throw no spanners. "I was outside at 6am smoking with my hood on, watching everyone come to work, freezing my ass off. That's how dedicated I am."

Then breakfast, another cigar, four hours of interviews and lunch in his room, Moby on the deck, steak on the plate. "I ordered it medium rare. I got it medium beyond." Did he finish it? "Oh yeah, every little bit. I am a carnivore." Then another cigar. He'd like to be puffing on one right now. At least three more will get sucked before he sleeps.

"They're excellent to study to," he explains, "you can put 'em down, pick 'em up – they keep you company while you're thinking. They're a good thinking man's tool." Eckhart creases; smile gummier than you might expect, blinding whites under wraps. At 45, his face still looks ripped from Mount Rushmore, chin dimple so deep you could keep your handbag in it. But the butter hair is closer-cropped these days, there's slightly more of a raptor aspect all round. These are not features looking for definition; big geek glasses add extra anyway.

He rocks back. "With cigars it's inherent in the mechanics that you're going to sit down for a long time. You plan your day around it. If you go to a cigar bar [he frequents one in Malibu] you're going to feel your bottom on the chair, the feet on the ground; you're going to have two or three coffees. We watch ball games together. We all have the same sort of crude sensibility and that's good because nobody feels out of place. And nobody else in their right mind will go in there because it stinks so bad."

Aaron Eckhart, as the US president (right), with Morgan Freeman in Olympus Has Fallen. Photograph: Millennium Films/Sportsphoto Ltd/Allstar

Are there ever any women there? "Let me tell you a secret about women and cigars …" He leans forward. Eckhart, it seems clear, wants to talk smoking. Or shopping. Or clubbing. Or the colour of his T-shirt ("Range Rover would call it Stornoway Grey"), or of my frock ("I would call that graphite"). He rolls up his jeans to show me his legs, freshly shaven for smoother cycling. He discusses waxing (he hasn't tried it), considers carefully whether his legs could get colder than those of other men. He is good at, and interested in, chat – for its own sake, and because of the difficulty, he thinks, of accurately capturing it both in print, and on film. He is, perhaps, not quite so energised by talking about his new movie, much as he loves it (and, for the record, it is great, shameless fun).

"So let's say I'm dating a girl and at first she'll have whatever concept of a cigar, and then let's say we are intimate and so we kiss and stuff like that. By a week or two she will be begging to smoke a cigar before we kiss. It's incredible! It's an habitude. It becomes part of the intimacy which is extremely weird because you'd think it would be revolting. I mean, if you polled the public whether it was worse to beat your grandmother or smoke a cigar I don't know who would win. People freaking hate cigars. And there's a stigma with it, that you're a certain kind of a person."

Eckhart began his screen career specialising in that kind, first as a smoothie shark who seduces and dumps a deaf colleague for kicks in Neil LaBute's In the Company of Men, then as a sexual narcissist in Your Friends and Neighbors – also by LaBute, a fellow Mormon (Eckhart is now lapsed) he met in college. But he made bad palatable in Thank You For Smoking, as a tobacco lobbyist supercharged with charm. That was in 2005, just after he himself quit booze and fags.

The reason, he says now, was the lack of control. Cigarettes encourage neuroses. "And by the third drink, you're all different people. With a cigar you're the same person from the start to the end. Whereas with drink and drugs you're either fighting or stealing each other's girlfriends or hating yourself. That was my alcohol experience." He laughs hard.

Eckhart's press suggests self-discipline is of paramount importance. He's spoken of the power he has to read body language and the advantage that gives him. Yet today he seems to be enjoying the strain on the leash. He can't really mind sailing close to the wind; after all, cigars might not leave you with a hangover, but they do come with consequences.

That, he says, is why he doesn't inhale. "Because I'm a serious athlete. I train every single day. I would never want to take smoke into my lungs. I know I'm getting some, but if I walked up to Piccadilly I'm getting a lot. But non-smokers think it's the most unhealthy, unattractive thing they've ever seen. I was in Australia one time, by a bridge, smoking a cigar and taking some pictures. These runners went by and this woman just had a visceral reaction, made a noise like 'Eeeerugaah'. And I said: 'Here I am standing in piss and vomit from last night and that's OK, but God forbid I smoke a cigar and have a little bit of peace.'" What did she say? "Oh, she was up the road by then. I let her know though. I'm not going to take that."

Eckhart does not smoke in Olympus Has Fallen. That's because he is playing his most apple-pie part yet: as the straight-shooting US president, an unwavering trouper, delightful to his wife, a sweetheart to his son, a hope for the nation. The ciggies are the preserve of the North Koreans who storm the White House, rack up a fantastic body count, then chain the president to a railing in the bunker. There he remains for most of the movie, trying not to spill secret nuclear codes. His only hope is Gerard Butler's wisecracking secret service maverick, Mike, who has managed to infiltrate the building and is picking off foreigners at a rate of knots.

Aaron Eckhart with Liana Liberato in The Expatriate.

That film is out next week; a fortnight back saw the release of The Expatriate, which has Eckhart doing the world-saving himself, rather than just watching from the sidelines. It's a less blockbusting prospect: set in Belgium, with Eckhart by far the biggest name, as a single dad CIA enforcer now on the run from his former employers. If his role in Olympus channels Harrison Ford – an elder statesman with a background in biffing – The Expatriate is an attempt to do what Taken did for Liam Neeson, recharging a still formidable force (watch out for a scene of shirtless Semtex manufacture).

Yes, agrees Eckhart, Olympus is the bigger hitter. "The Expatriate hasn't found its little place in the world." Why is that? "Well, honestly, I can't tell you. I mean I could tell you. I know the answer but I can't tell you. I'll just say some films resonate with people somehow. I enjoyed making it, but y'know it just .... it's all about the captain of the ship." And in this case that's who? "I couldn't possibly tell you. I couldn't possibly insinuate what you're about to extrapolate."

Eckhart likes to tease. This can encourage you to push. I was surprised, I say, that the president was quite so impotent in Olympus. In the climatic battle, he's knocked to the floor as Butler and the chief baddie slug it out. The fight progresses along conventional lines, with Butler winning, then their positions flipping and all seeming lost. We cut to Eckhart, whom you expect to stagger up and deliver the crucial blow, saving the world and rescuing the friend who, as has been foregrounded, taught him to box. But he doesn't. He just sits there and wheezes: "Come on, Mike".

Eckhart sighs at some length. "OK, hold on. Stop right there. You've really hit a nerve." He's not kidding. "How do you think I feel? You don't think I suggested that? You don't think the audience would have loved that? You don't think they'd have got out of their seats and fucking clapped?"

So why did it not happen? "Next question. You're giving me a fucking heart attack. Holy moly. This isn't your first rodeo. But yeah, it would have been good. It's pretty much set up for it, isn't it? What did I just say about The Expatriate? There are forces sometimes working against the betterment, I think, of the overall good."

OK: have action heroes had a crisis of masculinity lately? Eckhart riddles me again. "It's the way the business is going, and the world. You can't have any real enemies any more, you can't really say what you wanna say, you can't say what you mean. Political incorrectness, all that stuff. People are so hen ... tied? Picked? Anyway, they're so strapped down trying to be the perfect person that they've lost their identity. Gerry [Gerard Butler] lets it hang out in this movie. He lets people know who he is in real life. He's not afraid. People love that; it's attractive and it's what makes a movie star. I want men – or, not men – I just want my 70s action hero back. I want the code: 'You fuck with me and I fuck with you. You look at my girl and you're gonna get it.' I want families to go see these movies and them all to be inspired."

Aaron Eckhart and Helena Bonham Carter in Conversations with Other Women.

Eckhart's biggest payday was as Harvey Dent in The Dark Knight – another upstanding lawman (for most of it at least) – and another unimpeachably multiplex performance. LaBute aside, the further from the mainstream Eckhart wanders, the less successful he seems to be. His riskiest role was in Alan Ball's Towelhead, as an ex-soldier who sleeps with the 13-year-old daughter of his Palestinian neighbour. The film was iffily reviewed and took just $372,000 (£241,700); about the same as that in which he might be best, Conversations with Other Women, opposite Helena Bonham Carter. A certain disillusion would be forgivable. He's now trying to get his own directing and producing projects off the ground. "So I can get my own vision out. I feel I have a little bit more to say than just being tied up to a railing." Would he like an Oscar? "Shit yeah I'd like one. I'd like to deserve one. I'd like to go to the big show."

The day before we meet, he joined Twitter. "My agent made me. Because there's this crazy insane notion that it all matters for business. They'll tell me so-and-so has this many followers. I go: 'Yeah, but did it get him a job? Are they a better actor?' And they're like: 'No.' And I'm like: 'But everyone knows everything about him.'"

So why did you give in? "That's insulting. Churchill's turning over in his grave right now. Why did I give in. Well, I haven't really, have I? I have 13 followers and two tweets. I'm an old man. What do I need Twitter for? Mostly it's just to get chicks."

He's really that desperate? "Yes I am. I really am. I'm very shy. I know you don't believe it. It's true." Has he tried internet dating? He hasn't – people might recognise his picture. "And then they tweet it and everything. Like, I've had some weird experiences, not with dating sites … let's not go into that …"

Aaron Eckhart with Julia Roberts in Erin Brockovich. Photograph: Sportsphoto/Allstar/Cinetext Collection

Eckhart has been engaged, but has never married. Nor, more crucially, has he settled down with an onscreen partner. He's played against some big stars – Catherine Zeta-Jones, Nicole Kidman, Jennifer Aniston, Julia Roberts - but no one twice, never quite finding a Jennifer Lawrence to his Bradley Cooper. His great acting hero is Cary Grant; he's spoken in the past of his admiration for his enduring partnership with Katharine Hepburn. Does he feel he could have done with something similar?

"Yes. Yes. I would like that. Whenever I find an actress that is attractive and good and I like her, I would like to make several movies with her. Hepburn and Grant, Hepburn and Tracey, Hepburn and James Stewart … Maybe they were all just boinking Katharine Hepburn."

Our time's nearly up; his agent and the film's PR have come to retrieve him. There's just chance to ask about spiritual manifestation, the conviction you can action things by thought alone. "I don't believe it's easy. I believe it's a science and it's absolutely true. I believe I can heal myself. The other day I was chopping wood at my house at 5.30am and this bit whacked me in the eye, knocked me down, spilt my coffee. You just release everything; release the pain. And you're healing – boom, boom, boom, all of a sudden – pain's gone, eye's cool."

I'm smiling a bit; so are the other people in the room. Someone mentions that he might still be jetlagged. "Are you gonna call me crazy?" he grins. "Well, I guess if you're gonna be safe, what's the point?"

Later, I logon and become his 14th Twitter follower. On Monday he's up to 300 and dutifully posting updates about how nice the fans are in Rome, and how friendly South Africa is. By Tuesday, the account has been suspended. That cigar must have tasted especially sweet.

• Olympus Has Fallen is released in the UK on 17 April