This work aims to study different probes of Type Ia supernova (SN Ia) progenitors that have been suggested to be linked to the presence of circumstellar material (CSM). In particular, we have investigated, for the first time, the link between narrow blueshifted Na I D absorption profiles and the presence and strength of the broad high-velocity Ca II near-infrared triplet absorption features seen in SNe la around maximum light. With the probes exploring different distances from the SN; Na I D > 10(17) cm, high-velocity Ca TT features < 10(15) cm. For this, we have used a new intermediate-resolution X-shooter spectral sample of 15 SNe Ia. We do not identify a link between these two probes, implying either that, one (or both) is not physically related to the presence of CSM or that the occurrence of CSM at the distance explored by one probe is not linked to its presence at the distance probed by the other. However, the previously identified statistical excess in the presence of blueshifted (over redshifted) Na I D absorption is confirmed in this sample at high significance and is found to be stronger in SNe Ia hosted by late-type galaxies. This excess is difficult to explain as being from an interstellar-medium origin as has been suggested by some recent modelling, as such an origin is not expected to show a bias for blueshifted absorption. However, a circumstellar origin for these features also appears unsatisfactory based on our new results, given the lack of link between the two probes of CSM investigated.