The exospheres of Jupiter’s icy satellites Europa and Ganymede are mixtures of H2O, O2 and H2 and some minor constituents. H2O is released from the surface mainly through either direct sputtering, caused by the impact of energetic ions of Jupiter's magnetosphere, or sublimation. O2 and H2 are produced through chemical reactions among different products of H2O radiolytic decomposition. In the present study we investigate the generation of H2O and O2 exospheres around Europa and Ganymede, under the external conditions that are likely in the Jupiter's magnetospheric environment, applying the Europa Global model of Exospheric Outgoing Neutrals (EGEON, Plainaki et al., 2010; 2012; 2013). The variability of the spatial distribution of Europa’s O2 environment is investigated through the consideration of different configurations between the positions of Europa, Jupiter and the Sun. In order to model the magnetospheric ion precipitation to Ganymede’s surface we use as an input the global electric and magnetic fields from the global MHD model of Ganymede’s magnetosphere (Jia et al., 2009). We find that at Europa, at low altitudes, O2 is the dominant exospheric species and has the largest column density, equal to 1.5 1019/m2 at the dayside, along the Europa–Sun line, and 3 1018/m2 at the nightside along the same line; at higher altitudes (> 0.2 RE) H2O becomes the dominant species. The spatial distribution of Europa's O2 exosphere is explicitly time-variable due to the time-varying relative orientations of solar illumination and the incident plasma direction. The rate of supply of O-atoms to the Europa torus is estimated to be up to ~6.5 10^25/s. The EGEON results on the O2 column densities, at Europa, are consistent with the surplus of OI emission at the 900west longitude (leading hemisphere) observed by HST. At Ganymede, the maximum contribution to the exosphere comes from sublimated H2O and is located at small altitudes above the moon’s subsolar point. There is a close correspondence of the near surface spatial distribution of the directly sputtered-H2O molecules with the open closed magnetic field lines boundary that also agrees well with the Galileo magnetic field and plasma flow measurements. A slight asymmetry in the modeled H2O and O2 exosphere appears between sub-Jupiter and anti-Jupiter direction that seems to be consistent with the HST observations of Ganymede’s auroral emissions.