Operator quantum error correction (OQEC) extends the standard formalism of quantum error correction (QEC) to codes in which only a subsystem within a subspace of states is used to store information in a noise-resilient fashion. Motivated by recent work on approximate QEC, which makes it possible to construct subspace codes beyond the framework of perfect error correction, we investigate the problem of approximate operator quantum error correction (AOQEC). We demonstrate easily checkable sufficient conditions for the existence of approximate subsystem codes. Furthermore, we prove the efficacy of the transpose channel as a simple-to-construct recovery map that works nearly as well as the optimal recovery channel, with optimality defined in terms of worst-case fidelity over all code states. This work generalizes our earlier approach of using the transpose channel for approximate subspace correction to the case of approximate OQEC, thus bringing us closer to a full analytical understanding of approximate codes.