I can actually purr very realistically, and my cats appear to be interested when I do, although of course I haven't the slightest idea of what I'd be saying to them:)
If you want to try you need to imitate first the letter "R" as they do in France, which is roughly done by exhaling while keeping the upper rear part of the tongue in slight contact with the rear of the palate. It comes easy to me as I had this "R" pronunciation until about 17 when I started correcting it (with some of the family disapproving). Now do the same but not using any vocal cords, that is, only breath, then repeat also while inhaling. Keeping the mouth barely open helps a lot to give it its particular sound which with practice can become really close to a real cat purring.
Open it up until you’re just barely ‘purring’ on exhale then gently but quickly start to inhale (think of your breath as the bow on a violin). There will be a slight timbre change but cats have that too.
It sounds more like a dragon purring than a housecat but close enough.
I can do what OP was describing on inhale and exhale, and I have a small amount of didgeridoo experience, but I don't think circular breathing works here. Mind you, I'm not GOOD at circular breathing, but my understanding is that it works by using the cheeks to force air through the lips as you are inhaling.
The purring sound here though comes from too far back to make that work, I can't get air to pass through that part.
The sound comes from my throat. Circular breathing involves a continuous exhalation from the mouth forward, but the air in the throat still moves forward and backward, so it doesn't help.
As an australian child I was told is was a special australian thing so please do not puncture my bubble!
OK, with the obligatory jingoism out of the way: that's quite interesting. And having heard a long note held for some time, I'm embarrassed not to have realized this on my own.
My band teacher from middle/junior/high school tried to teach it at different points to any/all that was interested. There was one trumpet player and one sax player that could do it when I was in school.
Practice will improve the result. Same with whistling. I can whistle quite well on the inhale, now. It took me years to get right, simply because we don't make that motion with our face parts very often at all.
If you want to try you need to imitate first the letter "R" as they do in France, which is roughly done by exhaling while keeping the upper rear part of the tongue in slight contact with the rear of the palate. It comes easy to me as I had this "R" pronunciation until about 17 when I started correcting it (with some of the family disapproving). Now do the same but not using any vocal cords, that is, only breath, then repeat also while inhaling. Keeping the mouth barely open helps a lot to give it its particular sound which with practice can become really close to a real cat purring.