  The current generation of very-high-energy $gamma-$ray (VHE; E above 30 GeV)
detectors (MAGIC and H.E.S.S.) have recently demonstrated the ability to detect
the afterglow emission of GRBs. However, the GRB prompt emission, typically
observed in the 10 keV-10 MeV band, has so far remained undetected at higher
energies. Here, we investigate the perspectives of multi-messenger observations
to detect the prompt emission of short GRBs in VHE. Considering binary neutron
star mergers as progenitors of short GRBs, we evaluate the joint detection
efficiency of the Cherenkov Telescope Array (CTA) observing in synergy with the
third generation of gravitational wave detectors, such as the Einstein
Telescope (ET) and Cosmic Explorer (CE). In particular, we evaluate the
expected capabilities to detect and localize gravitational wave events in the
inspiral phase and to provide an early warning alert able to drive the VHE
search. We compute the amount of possible joint detections by considering
several observational strategies, and demonstrate that the sensitivities of CTA
make the detection of the VHE emission possible even if it is several orders
fainter than the one observed at 10 keV-10 MeV. We discuss the results in terms
of possible scenarios of production of VHE photons from binary neutron star
mergers by considering GRB prompt and afterglow emissions.
