  GRB 080913 and GRB 090423 are the most distant gamma-ray bursts (GRBs) known
to-date, with spectroscopically determined redshifts of z=6.7 and z=8.1,
respectively. The detection of bursts at this early epoch of the Universe
significantly constrains the nature of GRBs and their progenitors. We perform
population synthesis studies of the formation and evolution of early stars, and
calculate the resulting formation rates of short and long-duration GRBs at high
redshift. The peak of the GRB rate from Population II stars occurs at z=7 for a
model with efficient/fast mixing of metals, while it is found at z=3 for an
inefficient/slow metallicity evolution model. We show that in the redshift
range 6<z<10 essentially all GRBs originate from Population II stars,
regardless of metallicity evolution model. These stars (having small, but
non-zero metallicity) are the most likely progenitors for both long GRBs
(collapsars) and short GRBs (NS-NS or BH-NS mergers) at this epoch. Although
the predicted intrinsic rates of long and short GRBs are similar at these high
redshifts, observational selection effects lead to higher (factor of 10)
observed rates for long GRBs. We conclude that the two recently observed high-z
GRB events are most likely long GRBs originating from Population II collapsars.
