Even people who don’t care much for George Clooney (and there are some) would admit he handles this fame business with consummate skill. At a time when so many celebrities revel in being loud and embarrassing, falling drunkenly out of cars, flaunting their addictions, careering in and out of rehab, Clooney stays composed and cool in the eye of a media hurricane, maintaining his dignity and his sense of self.

This much was in evidence at the Venice Film Festival. He was promoting his new film, Michael Clayton, a sophisticated thriller about an amoral fixer in a corporate law firm. In Venice, he was easily the star most hotly in demand from both the public and the media: more so than Brad Pitt, Michael Caine, Richard Gere, Ewan McGregor or Charlize Theron. I watched him handling the frenzy with commendable calm. Clooney does his duty to his fan base by strolling casually along the red carpet in an elegant tuxedo, smiling, waving and swapping the occasional joke. In a large hotel suite, he gets a pack of journalists eating out of his hand by answering questions candidly, quotably, seriously or flippantly, as the question demands.

He seems utterly unruffled by it all. It made me wonder how he copes with being at the centre of this constant maelstrom of attention. Does it not ever irritate him? Is it wearying being “on” all the time? “I grew up around famous people in Cincinnati, Ohio,” he says. “In that world, my father was a big star. My aunt was a big star. And I saw how little it has to do with you. It’s all about luck.”

That much is true. His father, Nick, was the best-known television news presenter in the city, and was highly regarded throughout the industry in America for his integrity and hard-hitting accuracy. Rosemary Clooney, his aunt, had a huge pop career in the 1950s. When the hits dried up, she went into semi-retirement and battled psychiatric problems, emerging later as an accomplished jazz vocalist. (She died in 2002.)

“I saw my aunt become not famous,” Clooney recalls. “And it wasn’t like she became less of a singer along the way.” Still, neither Nick nor Rosemary commanded the degree of adulation George now enjoys or endures. So how does he keep a sense of perspective? It helps, he says, that he achieved fame a little later than some: his hit TV show ER, which had followed a string of flops and dud movies, came when he was 33.

“The problem with famous people in general is that they actually think they’re geniuses,” he says. “You get famous and you think: ‘Yes, of course I should be famous. I’ve earned it all.’ I got a TV show with a 10pm Thursday night time slot (in the US). It was a massive hit so now I get to do movies I want to do. It’s luck. You have to capitalise on it and be available when it happens, but it’s luck. So, once you understand that everything you’re doing is literally based on stars aligning, then you really don’t take it for granted. You enjoy it.”

Clooney claims he is at that point in his career where few things “rattle” him. “Do I get sick of being ‘on’? I’ve been around that my whole life. I know what it is.” But he’s a strong believer in presenting himself appropriately. When I last talked to him 18 months ago, one of his main complaints about George Bush (apart from his policies) was his public demeanour and lack of gravitas. “Bush was winking and chuckling with everybody the day before he’s sending kids off to war,” Clooney groused. “Just because you have an audience yelling and cheering doesn’t mean you giggle and wink. If you’re the leader of the free world, you have to be good at that stuff. But he didn’t understand the moment.”

Clooney doesn’t make that mistake. He deflects questions that are too trivial or veer too close to his private life with self-deprecating wisecracks. But when it comes to real issues or his work, he’s serious and businesslike. He’s proud of Michael Clayton and makes an eloquent case for it. In the film, which co-stars Tom Wilkinson and Tilda Swinton, Clooney plays a fatigued anti-hero: his marriage has folded, he has gambling debts and a business he started is on the skids. Yet, he finds it within himself to stand firm against a corrupt chemical company using his firm to deny culpability for poisoning people.

“He’s a man at the end of a rope,” Clooney says of Clayton. “He’s used up all his options. Everything has closed down for him.” Michael Clayton is a morally ambiguous hero in a story with a bittersweet ending. It recalls films Clooney admires from the 1960s and ’70s, such as Klute, All the President’s Men and Network.

He mentions that in preparing for Michael Clayton he watched Paul Newman playing another man at the end of his tether in The Verdict. That’s intriguing, because Clooney, like Newman (and, notably, Gregory Peck), is content to make surefire hit films for studios (in his case Ocean’s 11, 12 and 13) in order to make less commercial films that are important to him personally and politically.

Last year, he was a prime mover of the Oscar nominees Syriana and Good Night, and Good Luck films with a liberal agenda that caused hardline Rightwing commentators to proclaim him a traitor to his country. “I essentially make them for nothing,” Clooney says of such films. “I don’t put my own money in, but I don’t take any out. You can gamble. A lot of them don’t make their money back. But it’s my job to get those films made. I’m afraid of being 80 years old, and people saying to me, ‘So what was your legacy?’”

He seems intent on saving Hollywood from its baser instincts. The films Clooney and his producing partner Steven Soderbergh choose to make together are smart, provocative and intriguing in their diversity: Solaris, Out of Sight, Confessions of a Dangerous Mind, and last year’s brace of Oscar nominees.

One suspects that, to the dismay of conservative critics, he will persevere in making films with a liberal slant. “I don’t know whether they have any effect,” he says disarmingly, “but I’d hate to inhabit a world in which they didn’t get made.”

Above all, George Clooney will surely also continue to make celebrity look effortless, suave, even classy.

Michael Clayton is out on September 28.