  In this work we characterize the expected gravitational wave signal
detectable by the planned space-borne interferometer LISA and the proposed next
generation space-borne interferometer $\mu$Ares arising from a population of
primordial black holes orbiting Sgr A*, the super-massive black hole at the
Galactic center. Assuming that such objects indeed form the entire diffuse mass
allowed by the observed orbit of S2 in the Galactic center, under the
simplified assumption of circular orbits and monochromatic mass function, we
assess the expected signal in gravitational waves, either from resolved and
non-resolved sources. We estimate a small but non negligible chance of $\simeq$
10% of detecting one single 1 M$_{\odot}$ primordial black hole with LISA in a
10-year-long data stream, while the background signal due to unresolved sources
would essentially elude any reasonable chance of detection. On the contrary,
$\mu$Ares, with a $\simeq$ 3 orders-of-magnitude better sensitivity at $\simeq$
10$^{-5}$ Hz, would be able to resolve $\simeq$ 140 solar mass primordial black
holes in the same amount of time, while the unresolved background should be
observable with an integrated signal-to-noise ratio $\gtrsim$ 100. Allowing the
typical PBH mass to be in the range 0.01-10 M$_{\odot}$ would increase LISA
chance of detection to $\simeq$ 40% towards the lower limit of the mass
spectrum. In the case of $\mu$Ares, instead, we find a "sweet spot" just about
1 M$_{\odot}$, a mass for which the number of resolvable events is indeed
maximized.
