Very high-energy gamma -rays (VHE; E>100 GeV) have been detected from the direction of the Galactic Centre up to energies E>10 TeV. Up to now, the origin of this emission is unknown due to the limited positional accuracy of the observing instruments. One of the counterpart candidates is the super-massive black hole (SMBH) Sgr A*. If the VHE emission is produced within 1015 cm 1000 rG (rG=G M/c2 is the Schwarzschild radius) of the SMBH, a decrease of the VHE photon flux in the energy range 100-300 GeV is expected whenever an early type or giant star approaches the line of sight within milli-arcseconds (mas). The dimming of the flux is due to absorption by pair-production of the VHE photons in the soft photon field of the star, an effect we refer to as pair-production eclipse (PPE). Based upon the currently known orbits of stars in the inner arcsecond of the Galaxy we find that PPEs lead to a systematic dimming in the 100-300 GeV band at the level of a few per cent and lasts for several weeks. Since the PPE affects only a narrow energy band and is well correlated with the passage of the star, it can be clearly discriminated against other systematic or even source-intrinsic effects. While the effect is too small to be observable with the current generation of VHE detectors, upcoming high count-rate experiments like the Cherenkov telescope array (CTA) will be sufficiently sensitive. Measuring the temporal signature of the PPE bears the potential to locate the position and size of the VHE emitting region within the inner 1000 rG or in the case of a non-detection exclude the immediate environment of the SMBH as the site of gamma -ray production altogether.