We use an evolutionary turbulent model of Jupiter's subnebula to constrain the composition of ices incorporated in its regular icy satellites. We consider CO2, CO, CH4, N2, NH3, H2S, Ar, Kr and Xe as the major volatile species existing in the gas-phase of the solar nebula. All these volatile species, except CO2 which crystallized as a pure condensate, are assumed to be trapped by H2O to form hydrates or clathrate hydrates in the solar nebula. Once condensed, these ices were incorporated into the growing planetesimals produced in the feeding zone of proto-Jupiter. Some of these solids then flowed from the solar nebula to the subnebula, and may have been accreted by the forming Jovian regular satellites. We show that ices embedded in solids entering at early epochs into the Jovian subdisk were all vaporized. This leads us to consider two different scenarios of regular icy satellite formation in order to estimate the composition of the ices they contain. In the first scenario, icy satellites were accreted from planetesimals that have been produced in Jupiter's feeding zone without further vaporization, whereas, in the second scenario, icy satellites were accreted from planetesimals produced in the Jovian subnebula. In this latter case, we study the evolution of carbon and nitrogen gas-phase chemistries in the Jovian subnebula and we show that the conversions of N2 to NH3, of CO to CO2, and of CO to CH4 were all inhibited in the major part of the subdisk. Finally, we assess the mass abundances of the major volatile species with respect to H2O in the interiors of the Jovian regular icy satellites. Our results are then compatible with the detection of CO2 on the surfaces of Callisto and Ganymede and with the presence of NH3 envisaged in subsurface oceans within Ganymede and Callisto.