Accreting white dwarfs have long been considered as candidates for Type Ia supernova progenitors. With the discovery of luminous supersoft X-ray sources (LSXSs), a new class of white dwarf accretor has been proposed as a candidate progenitor system. This class, close-binary supersoft sources (CBSSs) is studied here. We combine a population synthesis analysis with detailed evolutionary calculations for individual systems. We find that there are uncertainties related to the possibility that some of these systems may experience a common envelope phase. Systems that do experience a common envelope that is fatal to further hydrogen accretion, can be expected to then undergo either C-O/He core mergers, or else a subsequent epoch of helium accretion onto a C-O white dwarf.The evolutionary calculations are essential to a meaningful computation of the rate of possible Type Ia explosions, and to reliable estimates of the uncertainties. In addition, the evolutionary calculations allow us to compute the amount of matter ejected prior to the explosion, and the history of local ionization due to the source. Some secondary characteristics of the supernova can therefore be predicted. Further, the possible existence of a pre-supernova nebula may allow surveys for nebulae in distant galaxies to successfully identify some progenitors.