The paper develops an asymptotic expansion method for forward-backward SDEs (FBSDEs) driven by the random Poisson measures with σ-finite compensators. The expansion is performed around the small-variance limit of the forward SDE and does not necessarily require a small size of the non-linearity in the BSDE’s driver, which was actually the case for the linearization method proposed by the current authors in a Brownian setup before. A semi-analytic solution technique, which only requires a system of ODEs (one is non-linear and the others are linear) to be solved, as well as its error estimate are provided. In the case of a finite jump measure with a bounded intensity, the method can also handle sate-dependent (and hence non-Poissonian) jumps, which are quite relevant for many practical applications. Based on the stability result, we also provide a rigorous justification to use arbitrarily smooth coefficients in FBSDEs for any approximation purpose whenever rather mild conditions are satisfied.