Type Ia Supernovae (SNe Ia) have high and homogeneous luminosities, making them an essential tool for measuring distances on a cosmic scale, useful to gauge the geometry and evolution of the Universe. However, the nature of the progenitor system of these explosions is still uncertain. The consensus view is that SNe Ia originate from an accreting carbon-oxygen white dwarf (WD) in a binary system, that undergoes a thermonuclear explosion as its mass approaches the critical Chandrasekhar limit. In the single degenerate (SD) model the mass donor is either a main sequence or an evolved star, while the double degenerate (DD) model involves a second WD as the mass donor. In the SD model non-accreted material blown away from the system prior to the explosion would remain as circumstellar material (CSM). Detection of such material in SN Ia spectra would support the SD model. Recently, claims for such detections were reported for four SN Ia events (SN2006X, SN2007le, SN 1999cl and SN2006dd), manifest as time variable Na I D absorption features. We report the analysis of the largest high resolution SN Ia spectra sample to date, consisting of 35 SN Ia events, obtained using the Keck HIRES and Magellan MIKE spectrographs. 22 of these events exhibit significant Na I D absorption. We report a statistical preference for blue-shifted absorption structures (similar to those seen in SN 2006X and SN 2007le) that is incompatible with observations of the Milky way absorption systems or an additional core collapse SN sample. This apparent asymmetry suggests that most of these features are intrinsic to the SNe themselves, supporting the SD model for the progenitor system of a large fraction of SNe Ia in nearby (z<0.06) spiral and lenticular galaxies.