It sounds like things are moving forward for KATY PERRY and JOHN MAYER.  A source says Katy has been having SLEEPOVERS at John's house, which is in a gated community in L.A., "It's very private and they've been sneaking in and out.  It has been going on for a while."

This past weekend they attended a party at the home of a Hollywood talent agent, and the "guests" included creatures from a wild animal preserve, including a kangaroo and an alligator.

A source says, "They were really cute.  They were standing near each other most of the night, playing with the animals outside.  [They] definitely enjoyed each other's company.  They almost acted like they've been a couple for a long time.  It definitely didn't seem like a new relationship."