We have considered 53 Type Ia supernovae (SNe Ia) observed by the Swift X-Ray
Telescope (XRT). None of the SNe Ia are individually detected at any time or in
stacked images. Using these data and assuming that the SNe Ia are a homogeneous
class of objects, we have calculated upper limits to the X- ray luminosity (0.2
- 10 keV) and mass-loss rate of L0.2-10 < 1.7\times1038 erg s-1 and M \cdot <
1.1\times10-6 M\odot yr-1 \times(vw)/(10 km s-1), respectively. The results
exclude massive or evolved stars as the companion objects in SNe Ia progenitor
systems, but allow the possibility of main sequence or small stars, along with
double degenerate systems consisting of two white dwarfs, consistent with
results obtained at other wavelengths (e.g., UV, radio) in other studies.