Aaron Eckhart is recounting a tale that, you feel, he’ll be telling his grandchildren in years to come – the time he first met Clint Eastwood. He spotted the 86-year-old actor-director a few years ago at the Golden Globes. “He was surrounded by [Steven] Spielberg and Brad Pitt and Angelina Jolie… I just made my way up to Clint, who doesn’t know me from a hole in the wall, and I said, ‘sir, I just hope to work with you one day.’ He looks at me, kinda like Dirty Harry, and he says, ‘yeah, we’ll see what’s shakin’.”

To be fair to Eckhart, he’s a tad more than “a hole in the wall”. His 20-year career has seen him directed by the likes of Sean Penn, in The Pledge (2001), Christopher Nolan in The Dark Knight (2008), and Brian De Palma The Black Dahlia (2006). But the 48-year-old can’t help but flush with the memory. “I was like, ‘oh fuck’. I was embarrassed, I was humiliated. I was in front of everybody. I walked away saying, ‘that’ll never happen’.” He takes a sip from his cup of tea. “And it happened.”

Eastwood’s Sully is one of two new movies Eckhart stars in. The other is Bleed For This, Ben Younger’s boxing drama. In both, he plays a real-life person. His character is Jeff Skiles, co-pilot on the infamous US Airways Flight 1549 – on 15 January 2009, the jet was forced to emergency-land on New York’s Hudson River after a flock of birds hit both engines shortly after take-off.

Rubbing shoulders: Aaron Eckhart as the legendary trainer Kevin Rooney in ‘Bleed For This’ – with Miles Teller as boxer Vinny Pazienza

“Playing a real person, if they’re alive, is daunting – because they have to live with the results of your efforts,” says Eckhart, who stars alongside Tom Hanks – Hanks plays Chelsey ‘Sully’ Sullenberger, the pilot who heroically landed the plane. “There are some people on Earth who are only going to know Sully as Tom Hanks. And Jeff Skiles, certainly, as me. Now that’s a big responsibility. Jeff still flies. I want people to come up to Jeff and go, ‘hey, that was an awesome movie’.”

While it’s a gripping account of this so-called “miracle on the Hudson”, when all 155 souls on board were saved, Eastwood intriguingly intertwines it with the aftermath. Sully and Skiles are forced to defend their actions amid an investigation by the National Transportation Safety Board. “He’s Abraham Lincoln,” says Eckhart, talking about the popular perception of the ultra-calm Sully. “He could be on Mount Rushmore. People would probably like him to run for President.”

If Eckhart lets Hanks do most of the heavy lifting in Sully, he risks more in Bleed For This. The story of Vinny Pazienza (played by Whiplash’s Miles Teller), the junior middleweight champ who almost lost his life after a terrifying car crash, Eckhart plays his boozy, balding trainer Kevin Rooney, the man who taught Mike Tyson how to fight. It’s an unflattering, unglamorous and almost unidentifiable performance from Eckhart, his square-jawed looks tucked away, his vocal patterns couched in a broad East Coast accent.

Right-hand man: Eckhart opposite Tom Hanks in the cockpit in ‘Sully’ (Warner) (Courtesy of Warner Bros. Enterta)

So much so, when Younger screened the film to Steven Soderbergh, who directed Eckhart in legal drama Erin Brockovich, it took Soderbergh 10 minutes to realise it was his former actor on screen. Now that’s what you call a disappearing act. “It makes me question that Steven doesn’t even know who I am,” says Eckhart. “But I’m not trying to stand out in that sense, other than being what the character demands me to be. And Kevin [Rooney] does look like that, so I felt like I was being true to the character.”

While some actors regularly disguise their looks, Eckhart never has. “Gaining weight was one thing. But doing this accent, that was scary for me because it put me in a vulnerable space,” he says. “You could very easily ruin the movie that way. You could be so annoying that…I believe that I could’ve been twice as extreme as I was. But if I were to do it again, I would go way bigger than I did. There’s no such thing as big or small. There’s only truthful.”

What it does emphasise is just how adaptable Eckhart is – perhaps more than he gives himself credit for. Since his breakthrough as the white-collar bully Chad in Neil Labute’s In The Company Of Men in 1997, he’s straddled leading man and character player (although disasters like I, Frankenstein didn’t exactly position him as an action star). While he tells me “I like being in the hero business”, he’s always been at his most interesting with morally ambiguous characters like his tobacco company spin-doctor in 2005’s Thank You For Smoking.

In reality, Eckhart is clean-living. He gave up smoking and drinking by self-hypnosis. His parents are Mormons and, if he’s no longer practising, he still takes ‘influence’ from Mormonism. He’s also one of Hollywood’s most eligible bachelors – one who avoids high-profile romances and tabloid scrutiny – and arguably one of the most well-travelled. Born in San Jose, California, where his father worked for a computer firm, his parents moved he and his two brothers to England when he was 13, settling down in Walton-on-Thames.

While the family later moved to Australia, Hawaii and Switzerland, landing in Surrey was a shock. “Oh my God – I thought I was going to Mars, if Mars was a prison,” he laughs. “That’s a 13 year-old’s view of going anywhere but California. I was being raised in the surf culture and then to be taken away from that [was hard]. I surfed a lot in Cornwall, though. I came to love England and Europe.” Even now, he speaks good French and talks about trips he took to Moscow (where he picked up a tea-drinking habit).

Having played another real-life person – football coach Darrell Royal in last year’s My All-American – Eckhart knows what he wants to do next. “I’m trying to find something to direct,” he says. “I want to see if I can impress myself. I’m 48, I’ve had a good education and I’ve learned so much from these great directors and actors that I want to see if I can tell a story.” As Eastwood might say, we’ll just have to see what’s shaking.