The actress, who ended a serious relationship last September with Jack White, has been seen frequently with the singer.
One source confirmed, "Renee and Damien have been seeing each other for a while and they’re both really happy."
"They share a mutual love of music and both are very deep thinkers. Renee is pretty intense and Damien is something of a poet so they’re the perfect match."
The actress recently visited the singer in Dublin and has been shopping for homes. Perhaps she would like to buy a new house to be close to her new Prince Charming.