Damon and Greengrass previously collaborated on the thrillers "The Bourne Supremacy" and "The Bourne Ultimatum."
"I think that they have a good way to do a prequel with someone else and basically make it about the Bourne identity, the actual identity," Damon told reporters in New York recently when asked if another "Bourne" movie is in his future, while promoting the Iraq-set action drama "Green Zone," which Greengrass also helmed.
"What any studio is interested in is just making it like an evergreen that can just go on and on and on. It never will with our character because he's going to resolve himself," Damon explained. "(My Jason Bourne's) resolved his issues now. He's gotten his memory back three times now. I don't think anybody wants to see me say, 'I don't remember' again. But I think what we could do is that you could do some movies with another actor, anyone, whether it's Ryan Gosling or Russell Crowe or Denzel Washington and he's Jason Bourne. Then at the end of his one movie or two movies or three movies, you see them getting ready to pass the identity on to me. So it just becomes like a 007. It becomes the name that they give this certain person who's uniquely positioned. So then if Paul and I come back and do a fourth one in 10 years, right, we'll pass it on to someone else and then the thing can kind of go on."
Co-starring Amy Ryan, Greg Kinnear and Brendan Gleeson, "Green Zone" is in theaters now. Damon plays a U.S. Army warrant officer looking for weapons of mass destruction in Baghdad in 2003.
"It's entertainment, but kind of based in the real world," Damon said. "So, if you start with the question of the weapons, which was the reason ostensibly that we went there, you start with a guy who's genuinely a good soldier and believes the weapons are there and you see him not find them and the next logical question he asks is: 'OK, but then what's going on? Why aren't they here?' And then that search kind of brings him here, into this world of the Green Zone. He's suddenly there and he's got very simple motives. He's there to win a war, to save lives. He's got very noble motives and suddenly finds himself in a very complicated world with these very powerful competing agendas and he's in the middle of it."
Celebrity Families of 2014 [PHOTOS]