In existence of mass, spacetime gets warped, this means time speeds up near the mass, so how can we measure the actual rate of flow of time?
There is no absolute time. Time always passes at the normal rate for you, your clock, and everything in your frame of reference no matter what your conditions. A distant observer considered to be in a different frame of motion, acceleration, and / or gravitational conditions will also perceive their own time perception and clock as running normally. Yet each of you may see time running at a different rate for the other. Each is equally valid, there is no privileged or absolute time.