  We study the radiative corrections to the stabilization of the complex
structure modulus $\tau$ in modular flavor symmetric models. We discuss the
possibility of obtaining the vacuum expectation value of $\tau$ in the vicinity
of the fixed point where residual symmetries remain unbroken. As concrete
examples, we analyze the 1-loop Coleman-Weinberg potential in the $A_4$ modular
flavor models. We show that the 1-loop correction may lead to the slight
deviation from the tree level result, which may realize a phenomenologically
preferred value of the complex structure modulus $\tau$ particularly when the
number of species contributing to the 1-loop correction is large enough.
