The singer, 32, and the actor, 48, have been spending time together, a source tells People.
People has reached out to both of their reps for comment.
The Oscar winner and the Avengers actor do share a love of music.
Renner wrote songs for the animated film Arctic Justice: Thunder Squad in which he also voiced the fox named Swifty. Last year, he told USA Today he had practiced “music since I was 12, I just didn’t want to do it as a business.