I have a circle that I want to export to use as an icon.

But the problem is when you export the circle it actually exports the background bounding box as well. I know the bounding box is transparent but it still causes issues on phones because they render shadows based on the square instead of a circle.

Welcome to InkscapeForum!

There's no way to have an image in the form of circle (although I think it would be a nice idea), in Inkscape, or anywhere else. But there must be some way to make an image into a phone icon, without having a shadow around the rectangle. Because clearly they exist. And as I've said, there's no way to have an image in the shape of a circle, in any program that I know of.

How did you export? I'm assuming you used File menu > Export Bitmap. But perhaps you used something else? For example, if you used Save As > Cairo png, a background would be applied to the image. Also, if you're using a Mac, and you pasted the circle, it could add a background (because of a known bug).

There's a bug in the phone software, all is well after a phone reboot