A recent reanalysis of Einstein data, and new ROSAT observations, have revealed the presence of at least two components in the X-ray spectra of X-ray faint early-type galaxies: a relatively hard component (kT > 1.5 keV), and a very soft component (kT approximately 0.2-0.3 keV). In this paper we address the problem of the nature of the very soft component, and whether it can be due to a hot interstellar medium (ISM), or is most likely originated by the collective emission of very soft stellar sources. To this purpose, hydrodynamical evolutionary sequences for the secular behavior of gas flows in ellipticals have been performed, varying the Type Ia supernovae rate of explosion, and the dark matter amount and distribution. The results are compared with the observational X-ray data: the average Einstein spectrum for six X-ray faint early-type galaxies (among which are NG-C 4365 and NGC 4697), and the spectrum obtained by the ROSAT pointed observation of NGC 4365. The very soft component could be entirely explained with a hot ISM only in galaxies such as NGC 4697, i.e., when the depth of the potential well-on which the average ISM temperature strongly depends-is quite shallow; in NGC 4365 a diffuse hot ISM would have a temperature larger than that of the very soft component, because of the deeper potential well. So, in NGC 4365 the softest contribution to the X-ray emission comes certainly from stellar sources. As stellar soft X-ray emitters, we consider late-type stellar coronae, supersoft sources such as those discovered by ROSAT in the Magellanic Clouds and M31, and RS CVn systems. All these candidates can be substantial contributors to the very soft emission, though none of them, taken separately, plausibly accounts entirely for its properties. We finally present a model for the X-ray emission of NGC 4365, to reproduce in detail the results of the ROSAT pointed observation, including PSPC spectrum and radial surface brightness distribution. The present data may suggest that the X-ray surface brightness is more extended than the optical profile. In this case, a straightforward explanation in terms of stellar sources could be not satisfactory. The available data can be better explained with three different contributions: a very soft component of stellar origin, a hard component from X-ray binaries, and a approximately 0.6 keV hot ISM. The latter can explain the extended X-ray surface brightness profile, if the galaxy has a dark-to-luminous mass ratio of 9, with the dark matter very broadly distributed, and a SN Ia explosion rate of approximately 0.6 the Tammann rate.
