The breaking of reciprocity is a topic of great interest in fundamental physics and optical information processing applications. We demonstrate nonreciprocal light transport in a quantum system of hot atoms by engineering the dissipative atomic reservoir. Our scheme is based on the phase-sensitive light transport in a multichannel photon-atom interaction configuration, where the phase of collective atomic excitations is tunable through external driving fields. Remarkably, we observe interchannel quantum correlations that originate from interactions with the judiciously engineered reservoir. The nonreciprocal transport in a quantum optical atomic system constitutes a new paradigm for atom-based nonreciprocal optics and offers opportunities for quantum simulations with coupled optical channels.