The Hurt Locker star will compete for the Best Actor honour at the upcoming Academy Awards, but Renner - who plays the piano and guitar - admits his dream has always been to star in something more music-based.
He tells People magazine, "I've always wanted to do some sort of musical. That's a dream for me. Maybe play Kurt Cobain."
And Renner could be in luck - The Messenger moviemaker Oren Moverman is reportedly in talks to take charge of a planned Cobain biopic.
The project, based on Charles R. Cross' 2001 book Heavier Than Heaven: A Biography of Kurt Cobain, has been delayed after Brothers screenwriter David Benioff pulled out as director.