Is there any pure digital phase shifter circuit?
Yes of course it's possible - any analogue filter can be taken into the Z domain and a digital equivalent constructed. This happens all the time. So if you have an analogue circuit that does what your need there are several methodologies that produce an equivalent digital "circuit" that can be implemented in MCUs, FPGAs etc..

Try looking up: -


Impulse invariance method
Bilinear transformation


As to whether your original analogue circuit is a pure delay, that is another matter. Anyway, I forgot that I wrote a paper on it and the front sheet should help you understand: -



It begins with a simple RC and converts it to an op-amp equivalent in position (2). Position (3) re-shapes it slightly and position (4) arrives at the finished digital filter.