Two EXOSAT observations of gamma-ray burst error boxes and their possibly associated optical transients are discussed. No point source was detected in either box. A comparison with Einstein data (Pizzichini et al., 1986) is made for one of them (November 19, 1978). The data are interpreted in the framework of neutron star models for gamma-ray bursters. Upper limits are derived for neutron star surface temperatures, accretion rates and surface densities of accretion disks. The lack of detectable X-ray emission imposes stringent constraints on accretion rates onto neutron stars, and there is a problem of consistency between EXOSAT and Einstein observations, if the data are interpreted in the framework of dense disk models.