  The neutrino magnetic moment operator clasps a tiny but non-zero value within
the standard model (SM) of particle physics and rather enhanced values in
various new physics models. This generation of the magnetic moment ($\mu_\nu$)
is through quantum loop corrections which can exhibit spin-flavor oscillations
in the presence of an external magnetic field. Also, several studies predict
the existence of a primordial magnetic field (PMF) in the early universe,
extending back to the era of Big Bang Nucleosynthesis (BBN) and before. The
recent NANOGrav measurement can be considered as a strong indication of the
presence of these PMFs. In this work, we consider the effect of the PMF on the
flux of relic neutrinos. For Dirac neutrinos, we show that half of the active
relic neutrinos can become sterile due to spin-flavor oscillations well before
becoming non-relativistic owing to the expansion of the Universe and also
before the timeline of the formation of galaxies and hence intergalactic
fields, subject to the constraints on the combined value of $\mu_\nu$ and the
cosmic magnetic field at the time of neutrino decoupling. For the upper limit
of PMF allowed by the BBN, this can be true even if the experimental bounds on
$\mu_{\nu}$ approaches a few times its SM value.
