The long gamma-ray burst (GRB) 100621A, at the time the brightest X-ray transient ever detected by Swift-XRT in the $0.3textrm{--}10$ keV range, has been observed with the H.E.S.S. imaging air Cherenkov telescope array, sensitive to gamma radiation in the very-high-energy (VHE, $>100$ GeV) regime. Due to its relatively small redshift of $zsim0.5$, the favourable position in the southern sky and the relatively short follow-up time ($<700 rm{s}$ after the satellite trigger) of the H.E.S.S. observations, this GRB could be within the sensitivity reach of the H.E.S.S. instrument. The analysis of the H.E.S.S. data shows no indication of emission and yields an integral flux upper limit above $sim$380 GeV of $4.2times10^{-12} rm cm^{-2}s^{-1}$ (95 % confidence level), assuming a simple Band function extension model. A comparison to a spectral-temporal model, normalised to the prompt flux at sub-MeV energies, constraints the existence of a temporally extended and strong additional hard power law, as has been observed in the other bright X-ray GRB 130427A. A comparison between the H.E.S.S. upper limit and the contemporaneous energy output in X-rays constrains the ratio between the X-ray and VHE gamma-ray fluxes to be greater than 0.4. This value is an important quantity for modelling the afterglow and can constrain leptonic emission scenarios, where leptons are responsible for the X-ray emission and might produce VHE gamma rays.
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