  Recently, a soft black-body component was observed in the early X-ray
afterglow of GRB 060218, which was interpreted as shock breakout from the thick
wind of the progenitor Wolf-Rayet (WR) star of the underlying Type Ic SN
2006aj. In this paper we present a simple model for computing the
characteristic quantities (including energy, temperature, and time-duration)
for the transient event from the shock breakout in Type Ibc supernovae produced
by the core-collapse of WR stars surrounded by dense winds. In contrast to the
case of a star without a strong wind, the shock breakout occurs in the wind
region rather than inside the star, caused by the large optical depth in the
wind. We find that, for the case of a WR star with a dense wind, the total
energy of the radiation generated by the supernova shock breakout is larger
than that in the case of the same star without a wind by a factor > 10. The
temperature can be either hotter or cooler, depending on the wind parameters.
The time-duration is larger caused by the increase in the effective radius of
the star due to the presence of a thick wind. Then, we apply the model to GRB
060218/SN 2006aj. We show that, to explain both the temperature and the total
energy of the black-body component observed in GRB 060218 by the shock
breakout, the progenitor WR star has to have an unrealistically large core
radius (the radius at optical depth of 20), larger than 100 Rsun. In spite of
this disappointing result, our model is expected to have important applications
to the observations on Type Ibc supernovae in which the detection of shock
breakout will provide important clues to the progenitors of SNe Ibc.
