In studies on intermediate- and high-resolution spectra of Type Ia supernovae (SNe Ia), some objects exhibit narrow Na I D absorptions often blueshifted with respect to the rest wavelength within the host galaxy. The absence of these in other SNe Ia may reflect that the explosions have different progenitors: blueshifted Na I D features might be explained by the outflows of 'single-degenerate' systems (binaries of a white dwarf with a non-degenerate companion). In this work, we search for systematic differences among SNe Ia for which the Na I D characteristics have been clearly established in previous studies. We perform an analysis of the chemical abundances in the outer ejecta of 13 'spectroscopically normal' SNe Ia (five of which show blueshifted Na lines), modelling time series of photospheric spectra with a radiative-transfer code. We find only moderate differences between 'blueshifted-Na', 'redshifted-Na' and 'no-Na' SNe Ia, so that we can neither conclusively confirm a 'one-scenario' nor a 'two-scenario' theory for normal SNe Ia. Yet, some of the trends we see should be further studied using larger observed samples: models for blueshifted-Na SNe tend to show higher photospheric velocities than no-Na SNe, corresponding to a higher opacity of the envelope. Consistently, blueshifted-Na SNe show hints of a somewhat larger iron-group content in the outer layers with respect to the no-Na subsample (and also to the redshifted-Na subsample). This agrees with earlier work where it was found that the light curves of no-Na SNe - often appearing in elliptical galaxies - are narrower, that is, decline more rapidly.