Recently the first author has proved the asymptotic completeness for $N$-body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is nonzero by obtaining propagation estimates for the physical propagator. However, in his work, it is needed that potentials under consideration are sufficiently smooth. In this paper, when $N=2$, we prove the asymptotic completeness of (modified) wave operators under the assumption that the potential has the local singularity of type $|x|^{-1+\epsilon}$ when $d\ge3$ and $0<\epsilon<1$, where $d$ is the space dimension. We also discuss the modifiers in the position representation, which are used in the definition of modified wave operators in the long-range case.