It is generally believed that cosmological gamma-ray bursts (GRBs) are produced by the deceleration of relativistic objects with Gamma greater, similar100. We study the possibility that some GRBs are produced along with relativistic matter ejected from supernovae. In this model, it is quite likely that the matter has to travel through the progenitor's thick envelope before generating GRBs. Under the assumption that the ejected matter is described as a single collective piece of matter, we obtain constraints on the matter having Gamma greater, similar100 at the breakout of the progenitor. One advantage of considering this type of model is that the expected GRB energy is sufficiently large, in contrast to the GRB-generation model of the shock breakout in the energetic supernova explosion. We find that, in general, the cross section of the matter has to be very small compared with the progenitor's radius, and thus the matter has to be bullet-like (or jetlike) rather than shell-like.