The fluxes of the inverse Compton gamma-rays expected from synchrotron X-ray nebulae are calculated and the observability of this radiation is discussed. The main emphasis is given to the pulsar driven nebulae (plerions), although the results and conclusions are equally applicable to the extended non-thermal X-ray sources produced by shock-accelerated electrons in the shell-type supernovae remnants. The existence of the non-thermal (synchrotron) component of X-radiation in these objects implies an effective acceleration of electrons up to energies E-e similar to 100B(-5)(1/2) epsilon(keV) TeV (B-5=B/10(-5) G; epsilon(keV)=epsilon/1keV). The inverse Compton scattering of the same electrons on the ambient photon fields may result in observable TeV gamma-radiation as well. The 2.7-K microwave background radiation is, as a rule, the dominant target photon field for production of gamma-rays. This provides a direct relation, for the given magnetic field, between the typical energies of the synchrotron (epsilon) and inverse Compton (E) photons produced by the same electrons: epsilon(keV)similar or equal to 0.07(E/1 TeV)B-5. The ratio of relevant energy fluxes at these energies is about f(gamma)(greater than or equal to E)/f(x)(greater than or equal to epsilon)similar or equal to 0.1 B(-5)(-2)xi, where the f(x)(greater than or equal to E) is the energy flux of soft X-rays corrected for absorption, and the factor xi greater than or equal to 1 is introduced in order to take into account possible differences in the source sizes responsible for the fluxes observed by X-ray and gamma-ray detectors. Since the fluxes of X-ray nebulae with angular size less than a few arcmin are typically at the level of f(x) less than or equal to 10(-11) erg cm(-2) s(-1), then the detectability of these objects in TeV gamma-rays, by current atmospheric Cherenkov telescopes with sensitivities a few times 10(-12) erg cm(-2) s(-1), would significantly depend on the ambient magnetic field. In particular, the gamma-ray observability of these X-ray nebulae becomes problematic even for the lowest possible magnetic field, i.e. B similar to B(ISM)similar or equal to 3-5 mu G. Otherwise, the detection of gamma-rays from such sources would require xi>>1, which implies that in fact the relativistic electrons occupy a significantly larger region around the accelerator than the greater than or equal to 1 arcmin X-ray nebulae resolved by the ROSAT and ASCA satellites. We argue that the invocation of such an hypothesis allows us to explain satisfactorily the flux of TeV gamma-rays detected from the direction of the recently discovered faint X-ray nebula around the pulsar PSR B1706-44.
