  We have calculated the temperature dependence of shear $\eta$ and bulk
$\zeta$ viscosities of quark matter due to quark-meson fluctuations. The quark
thermal width originating from quantum fluctuations of quark-$\pi$ and
quark-$\sigma$ loops at finite temperature is calculated with the formalism of
real-time thermal field theory. Temperature-dependent constituent-quark and
meson masses, and quark-meson couplings are obtained in the Nambu--Jona-Lasinio
model. We found a non-trivial influence of the temperature-dependent masses and
couplings on the Landau-cut structure of the quark self-energy. Our results for
the ratios $\eta/s$ and $\zeta/s$, where $s$ is the entropy density (also
determined in the Nambu--Jona-Lasinio model in the quasi-particle
approximation), are in fair agreement with results of the literature obtained
from different models and techniques. In particular, our result for $\eta/s$
has a minimum very close to the conjectured AdS/CFT lower bound, $\eta/s =
1/4\pi$.
