The explosion of a Type Ia supernova, SN 2011ef, in the nearby Pinwheel galaxy (M101 at 6.4 Mpc) provides an opportunity to study pre-explosion images and search for the progenitor, which should consist of a white dwarf (WD), possibly surrounded by an accretion disk, in orbit with another star. We report on our use of deep Chandra observations to limit the luminosity and temperature of the pre-explosion white dwarf (WD). It is found that if the spectrum was a blackbody, then WDs of highest possible temperatures and luminosities are excluded but, even if the WD was emitting at the Eddington luminosity, values of kT less than roughly 60 eV are permitted. This allows the progenitor to be an accreting nuclear-burning WD with an expanded photosphere. Pre-SN HST observations were used to derive a lower limit of about 10 eV for the expanded photosphere. Li et al.\, (2011) have already ruled out the possibility of a giant donor. We consider the combined emission from the WD, disk, and donor, and find that even the combined emission from a bright subgiant, WD and disk would not likely have been observed prior to explosion, and neither would some local candidates for the nuclear-burning WD model. ; Comment: 14 pages, 3 figures, 2 tables, submitted to ApJ Letters