  Accumulating evidence indicates that some of ultra-luminous X-ray sources
(ULXs) are intermediate mass black holes (IMBHs), but the formation process of
IMBHs is unknown. One possibility is that they were formed as remnants of
population III (Pop III) stars, but it has been thought that the probability of
being an ULX is too low for IMBHs distributed in galactic haloes to account for
the observed number of ULXs. Here we show that the number of ULXs can be
explained by such halo IMBHs passing through a dense molecular cloud, if Pop
III star formation is very efficient as recently suggested by the excess of the
cosmic near-infrared background radiation that cannot be accounted for by
normal galaxy populations. We calculate the luminosity function of X-ray
sources in our scenario and find that it is consistent with observed data. Our
scenario can explain that ULXs are preferentially found at outskirts of large
gas concentrations in star forming regions. A few important physical effects
are pointed out and discussed, including gas dynamical friction, radiative
efficiency of accretion flow, and radiative feedback to ambient medium. ULXs
could last for ~10^{5-6} yr to emit a total energy of ~10^{53} erg, which is
sufficient to power the ionized expanding nebulae found by optical
observations.
