Our main aim is to test the non-variability of the radial velocity (RV) of a sample of 2351 standard stars used for wavelength calibration of the Radial Velocity Spectrometer (RVS) instrument onboard Gaia . In this paper, we present the spectroscopic analysis of these stars with the determination of their physical parameters by matching observed and synthetic spectra. Weestimate the offset between different instruments after determining the shift between measured and archived RVs since the instrument pipelines use various numerical masks. Through the confirmation of the stability of the target RVs, we find 68 stars with a long-term variation having an acceleration that exceeds 10m/s/yr. This suggests a barycentric reflex motion caused by a companion. As activity phenomena may be the source of periodic and trend-like RV variations in stars with putative planetary companions, we analysed various activity indicators in order to check their correlations to the RV changes. Among the trend stars, 18 have a trend model scatter greater than 100m/s^ over a time span from 10 to 12 yr. We also confirm that six stars with known substellar companions have a total model scatter, 3{sigma}, exceeding the threshold set by Gaia, that is, 300m/s. In addition, TYC8963-01543-1, an SB2 star, has data scatter {sigma}=176.6m/s. Four more other stars are revealed to be variable after combining data from different instruments. Despite the presence of low-amplitude changes, a very large fraction of our sample (98.8 per cent) appears suitable as RV calibrators for Gaia RVS.