  The upcoming discoveries of pulsars orbiting the center of the Milky Way will
present unparalleled opportunities to examine the causal structure of the
spacetime geometry of Sagittarius A*. In this paper, we investigate the fully
relativistic propagation time delay of pulsar signals in the
Joshi-Malafarina-Narayan (JMN-1) and Janis-Newman-Winicour (JNW) spacetimes.
This delay arises basically from the spacetime curvatures in the vicinity of
these ultra-compact objects, induced by the intense gravitational field near
the Galactic Center (GC). Using the principles of gravitational lensing, we
compute the arrival time of photons originating from a pulsar in orbit around
the GC. To validate our approach, we compare our time delay analysis of the
Schwarzschild black hole with the corresponding delay in the post-Newtonian
framework. Subsequently, we find that the propagation time of pulsar signal is
greater and lesser for the given horizon-less ultra-compact objects for direct
and indirect propagation respectively. Therefore, our results suggest quite
significant propagation time delay differences in JMN-1 and JNW spacetimes,
when compared to the Schwarzschild black hole case. This can be inferred as a
possible distinguishing feature for these ultra-compact objects' geometries.
