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, $eta/s = 1/4pi$.
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