  Recent numerical simulations have suggested the probability of a fraction of
the primordial stars to be ejected from the cluster of their origin. We explore
the possibility that some of these can remain on the main sequence till the
present epoch. We develop a semi-analytical model guided by results of
cosmological simulations to study the mass accretion by these protostars as a
function of the original stellar mass, and other parameters such as angular
momentum and gravitational drag due to ambient gas. We also explore whether
some of the protostars remain sufficiently low-mass and long-lived to survive
to the present day. This requires that the protostars are ejected from the star
forming region while their mass is less than $0.8\,M_{\odot}$. Assuming the
protostars gain mass via the spherical Bondi-Hoyle accretion from the ambient
medium, we show that Pop III protostars that initially form within a certain
range of mass and are ejected with velocity larger than the escape velocity may
survive to the present day on the main sequence. Thus, they may even be found
in our Milky Way or its satellites. Our calculations also reveal that
protostars that do not get ejected from the parent gas cloud accrete large
amount of gas. We predict that these can become massive enough to be
progenitors of black holes.
