The Hurt Locker star has replaced Damon as the lead actor in The Bourne Legacy, where he plays a character called Aaron Cross.
Renner told ShortList: "They keep Jason Bourne alive, so who knows where they'll actually go with it.
"This one still has to be successful, but maybe eventually Matt Damon and I could do one together. That would be kick-ass. I love Matt."
He added of the upcoming film: "We've kept the name, the style, tone and format of what those Bourne movies are.
"It's a new [shady government training] programme like Treadstone with a new set of spies designed to do what they do."
Renner continued: "We're filming in Calgary until Christmas then go to the Philippines for a couple of months to finish it off.
"Everything about it is the same. The way it's shot, the pace, the ticking clock and the very visceral fights. It's down, dirty and gritty."
Damon has said that he would like to reunite with director Paul Greengrass for a new Bourne movie in the future and is "keeping hope" that it will happen.
Renner previously said that he would like to work with Damon because he is a "great guy" who has "always been so complimentary to me from the moment I met him".
Watch Jeremy Renner in the Mission Impossible: Ghost Protocol trailer below: