Two outer bounds on the admissible source region for broadcast channels with correlated sources are presented: the first one is strictly tighter than the existing outer bound by Gohari and Anantharam, while the second one provides a complete characterization of the admissible source region in the case where the two sources are conditionally independent given the common part. These outer bounds are deduced from the general necessary conditions established for the lossy source broadcast problem via suitable comparisons between the virtual broadcast channel (induced by the source and the reconstructions) and the physical broadcast channel.