He adds, "I've known Matt over the past couple of years and we spoke about it for literally 30 seconds."
As the fourth chapter in the series of novels created by Robert Ludlum, Damon was initially attached to the project but cancelled when director Paul Greengrass, who directed the last two Bourne films, decided he wouldn't continue. Damon wouldn't sign on without him.
With a reported budget of $90 million, The Bourne Legacy is written and directed by Tony Gilroy (Michael Clayton), who co-wrote the screenplays for the previous Bourne films based on Robert Ludlum's original page-turners. It also stars Edward Norton and Rachel Weisz as Cross' wife. Returning to their roles in the Treadstone programme are Albert Finney, Joan Allen, David Strathairn, and Scott Glenn.
"It's a different world in this Bourne universe than you've seen previously,"says Renner. "The obvious disparity is that Jason Bourne didn't know who he was and that's what he was always trying to figure out. This guy, Aaron Cross, knows exactly who he is and how he's involved, and he wants to be involved."
Renner says he's feeling no pressure taking on the Bourne mantle.
"It's no different than any other job I've done. I'm a fan of the series but it has no bearing on how I do my job. I went to work every day and tried to make it as authentic as I possibly could. I feel as an actor I fit into that wheelhouse of what the franchise represents."
And he has a theory as to their popularity.
"I think people love them because they feel authentic. The stories are visceral and immersive and I think there's a form of fantasy when something feels so real and you care for the characters. That's why I think it does well.
"And also, they're smart, like a game of speed chess, essentially. They're thinking action movies and have elements for a lot of broad audiences."
Renner is amused that after The Avengers, in which he played the role of the super-archer Hawkeye, he's got a fanclub, some of whom can't yet see his other films.
"I never thought I'd have fans who were little kids. I've always done movies that were kind of intense and heavy in content, and to do a movie where you have little kids running around a backyard with bow and arrows, shooting each other, having a lot of fun, is really cool. The fact that there is an action figure of Hawkeye from The Avengers is kind of creepy but also great."
In real life, he couldn't be more different than most of the roles he plays.
"I'm definitely not a tough guy, and I think I'd make a terrible spy."
Who: Jeremy Renner as Aaron Cross
What: The Bourne Legacy
When: Opens next Thursday
- TimeOut