The 43-year-old actor detailed the possibility in an interview with Collider published Monday.
"There are rumblings of Captain America 3," he confirmed. "I don't know what's going to happen there. I've got to find the time, man. I'm trying to stay home and I can't."
Renner appeared as the master archer in The Avengers (2012), and will return for sequel Avengers: Age of Ultron in 2015. Hawkeye and Black Widow (Scarlett Johansson) are the only Avengers to not star in standalone movies; Iron Man (Robert Downey Jr.), Thor (Chris Hemsworth) and Captain America (Chris Evans) have their own series, and Hulk (Mark Ruffalo) was previously brought to life by Eric Bana and Edward Norton.
Captain America 3 is scheduled for release May 6, 2012. Renner will star in Kill the Messenger in October, and will appear in Avengers: Age of Ultron and Mission: Impossible 5 in 2015.