default FBO resource usage

Is it possible to use system default fbo depth renderbuffer in the user’s own frame buffer object or in other way - attach custom depth renderbuffer to default fbo?
I want to eliminate redundant data copying from the custom fbo to system fbo.

No I don’t think so though I am not well aware of the latest spec. The default attachment object type is NONE according to the spec thus does not correspond to any of the window system provided buffers.

Even if it was possible, this is pretty useless since buffers size between application provided framebuffers and window system provided framebuffer in most cases never matches.

And you don’t attach anything to the window system provided framebuffer, this is the purpose of custom framebuffers (fbo).