The "Cafe Society" actress was spotted holding hands and kissing the singer, whose real name is Annie Clark, in New York on Sunday evening and the pair, who have been seen together a number of times in recent weeks, are said to have a "definite spark" between them, reported Female First.
"Kristen and Annie have been really enjoying getting to know each other, there's a definite spark there. Kristen thinks Annie is a genius and Annie is equally taken with her," a source said.
"Kristen and Alicia haven't officially split but it's not the first time Kristen has put her on the back-burner," the source said.
Meanwhile, St Vincent was previously in a relationship with supermodel-and-actress Cara Delevingne.