A large number of early-type galaxies are now known to possess blue and red subpopulations of globular clusters. We have compiled a data base of 28 such galaxies exhibiting bimodal globular cluster colour distributions. After converting to a common V-I colour system, we investigate correlations between the mean colour of the blue and red subpopulations with galaxy velocity dispersion. We support previous claims that the mean colours of the blue globular clusters are unrelated to their host galaxy. They must have formed rather independently of the galaxy potential they now inhabit. The mean blue colour is similar to that for halo globular clusters in our Galaxy and M31. The red globular clusters, on the other hand, reveal a strong correlation with galaxy velocity dispersion. Furthermore, in well-studied galaxies the red subpopulation has similar, and possibly identical, colours to the galaxy halo stars. Our results indicate an intimate link between the red globular clusters and the host galaxy; they share a common formation history. A natural explanation for these trends would be the formation of the red globular clusters during galaxy collapse.