The first- and second-order optimum achievable rates in the source resolvability problem are considered for general sources. In the literature, the achievable rates in the resolvability problem with respect to the variational distance as well as the normalized Kullback-Leibler (KL) divergence have already been analyzed. On the other hand, in this study we consider the source resolvability problem with respect to (unnormalized) KL divergence and derive general formulas of the first- and second-order optimum achievable rates. Relationships with other problems in information theory have also been discussed.