  We assess the feasibility of detecting the compact object remnants from
Population III (Pop III) stars in nearby dense star clusters, where they become
luminous again as X-ray binaries (XRBs) and tidal disruption events (TDEs) via
strong tidal encounters. Analytically modelling the formation of Pop III stars,
coupled with a top-heavy initial mass function predicted by numerical
simulations, we derive the number of (active) Pop III XRBs and TDEs in the
present-day Milky Way (MW) nuclear star cluster as $\sim 0.06-0.3$ and
$\lesssim 4\times 10^{-6}$, rendering any detection unlikely. The detection
probability, however, can be significantly boosted when surveying all massive
star clusters from the MW and neighboring galaxy clusters. Specifically, we
predict $\sim 1.5-6.5$ and $\sim 40-2800$ active Pop III XRBs in the MW and the
Virgo cluster, respectively. Our Pop III XRBs are dominated ($\sim 99\%$) by
black holes with a typical mass and luminosity of $\sim 45$ $\rm M_{\odot}$ and
$\sim 10^{36}$ $\rm erg\ s^{-1}$. Deep surveys of nearby ($\lesssim 30-300$
$\rm Mpc$) galaxy clusters for such Pop III XRBs are well within reach of
next-generation X-ray telescopes, such as ATHENA and LYNX.
