  We study the impact of the extra density fluctuations induced by primordial
magnetic fields on the reionization history in the redshift range: $6 < z <
10$. We perform a comprehensive MCMC physical analysis allowing the variation
of parameters related to primordial magnetic fields (strength, $B_0$, and
power-spectrum index $n_{\scriptscriptstyle \rm B}$), reionization, and
$\Lambda$CDM cosmological model. We find that magnetic field strengths in the
range: $B_0 \simeq 0.05{-}0.3$ nG (for nearly scale-free power spectra) can
significantly alter the reionization history in the above redshift range and
can relieve the tension between the WMAP and quasar absorption spectra data.
Our analysis puts upper-limits on the magnetic field strength $B_0 < 0.358,
0.120, 0.059$ nG (95 % c.l.) for $n_{\scriptscriptstyle \rm B} = -2.95, -2.9,
-2.85$, respectively. These represent the strongest magnetic field constraints
among those available from other cosmological observables.
