Though the stars have stayed quiet on rumblings that they're a romantic item, Simpson, 29, writes on Twitter that the pair are collaborating on their music.
"I am blessed," the pop singer says, linking to two photographs of her with Corgan, 42, in the recording studio. She adds that it's "fun to be lost in laughter."
A source tells PEOPLE that the unlikely couple have been dating for about four months – longer than has been reported in the media – and that they met in a neighborhood where one was living and the other was staying with a friend.
"He got her number and starting calling her," the source says. "He thought she was cute ... adorable ... and she got a little crush."
Last month, when news of the possible relationship surfaced after the singers were spotted together in New York City, a source told PEOPLE that Simpson and Corgan were dating but "taking it slow."
Tuesday, a source close to Simpson said, "They are still close and still really good friends. They are still involved in each other's lives. They are still hanging out," adding, "I don't know if it's romantic anymore."
Reporting by K.C. BAKER and JENNIFER GARCIA