  Precursor emission has been observed in a non-negligible fraction of
gamma-ray bursts.The time gap between the precursor and the main burst extends
in some case up to hundreds of seconds, such as in GRB041219A, GRB050820A and
GRB060124. Both the origin of the precursor and the large value of the time gap
are controversial. Here we investigate the maximum possible time gaps arising
from the jet propagation inside the progenitor star, in models which assume
that the precursor is produced by the jet bow shock or the cocoon breaking out
of the progenitor. Due to the pressure drop ahead of the jet head after it
reaches the stellar surface, a rarefaction wave propagates back into the jet at
the sound speed, which re-accelerates the jet to a relativistic velocity and
therefore limits the gap period to within about ten seconds. This scenario
therefore cannot explain gaps which are hundreds of seconds long. Instead, we
ascribe such long time gaps to the behavior of the central engine, and suggest
a fallback collapsar scenario for these bursts. In this scenario, the precursor
is produced by a weak jet formed during the initial core collapse, possibly
related to MHD processes associated with a short-lived proto-neutron star,
while the main burst is produced by a stronger jet fed by fallback accretion
onto the black hole resulting from the collapse of the neutron star. We have
examined the propagation times of the weak precursor jet through the stellar
progenitor. We find that the initial weak jet can break out of the progenitor
in a time less than ten seconds (a typical precursor duration) provided that it
has a moderately high relativistic Lorentz factor \Gamma>=10 (abridged).
