Collisions were suggested to potentially play a role in the formation of massive stars in present day clusters, and have likely been relevant during the formation of massive stars and intermediate mass black holes within the first star clusters. In the early Universe, the first stellar clusters were particularly dense, as fragmentation typically only occurred at densities above 10(9) cm(-3), and the radii of the protostars were enhanced as a result of larger accretion rates, suggesting a potentially more relevant role of stellar collisions. We present here a detailed parameter study to assess how the number of collisions and the mass growth of the most massive object depend on the properties of the cluster. We also characterize the time evolution with three effective parameters: the time when most collisions occur, the duration of the collisions period, and the normalization required to obtain the total number of collisions. We apply our results to typical Population III (Pop. III) clusters of about 1000 M-circle dot, finding that a moderate enhancement of the mass of the most massive star by a factor of a few can be expected. For more massive Pop. III clusters as expected in the first atomic cooling halos, we expect a more significant enhancement by a factor of 15-32. We therefore conclude that collisions in massive Pop. III clusters were likely relevant to form the first intermediate mass black holes.