  We show that the dynamical quark mass in effective nonlocal models can vanish
at zero virtuality of the quark as $M(p^2)\propto p^2$. Our arguments follow
from the constrained-instanton model of the QCD vacuum and from QCD sum rules
calculations with nonlocal condensates. The discussed models also lead to
analyticity of $M(p^2)$ in the vicinity of zero.
