SiO2 fibers doped with rare-earth-doped nanocrystals are promising to overcome the strong SiO2 multi-phonon quenching and could yield novel laser gain materials. So far, no attention has been paid to the question how well the nanocrystals can suppress the quenching depending on the properties of the SiO2 host glass. Here, a novel analytical model was applied to study the impact of the glass purity and composition on the quenching efficiency. Only a few experimentally accessible material and design parameters are required to compute the average quenching rate inside the nanocrystals. It is demonstrated that sufficiently low levels of quenching can only be expected for SiO2 free of impurities or dopants that increase the multi-phonon absorption. This indicates that high-purity aluminosilicate glasses, in contrast to phosphosilicate and borosilicate glasses, are ideal hosts.