The 'Wide Awake' singer has been dating Ackroyd since April and the guitarist recently described their romance as "the real deal".
Perry appeared less committal when asked about Ackroyd in a recent chat with USA Today.
"That's not even appropriate to label. There are times I go out and meet people and flirt, but it's not really appropriate to have anything serious," she explained.
Perry went on to stress that she does hope to find love again following her divorce from Russell Brand.
"I have a second chance on life. To get maybe a better situation. For me, right now, I'm totally not looking. I'm just looking to heal. And I'm on that path. I'm not down on love," she commented.
The 27-year-old star previous commented that splitting from Brand taught her valuable lessons about life.
Her documentary Katy Perry: Part of Me is now playing in the US and the UK.
Watch a trailer for Katy Perry: Part of Me below: