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.”
“It’s not like I’m dropping an album or anything like that, [but it’s] a different type of music than I’d typically write, so I love that challenge,” he said.
“It just didn’t work out. Relationships sometimes end,” a source told PEOPLE at the time, adding that the pair split “a bit ago. There’s no long dramatic story.”
Since then, the singer has faced speculation about her chemistry with her A Star Is Born costar and director Bradley Cooper after a passionate Oscars performance.
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She told Jimmy Kimmel during an appearance on Jimmy Kimmel Live that there was nothing more to their relationship than that of friends.
“Yes, people saw love and guess what? That’s what we wanted you to see,” she said. “[‘Shallow’] is a love song, A Star is Born is a love story. It was so important to both of us that we were connected the entire time… When you’re singing love songs, that’s what you want people to feel.”
“Rumors I’m pregnant? Yeah, I’m pregnant with #LG6,” she tweeted.
Renner has a daughter, Ava, 5, with ex-wife Sonni Pacheco.