Spin transport through magnetic insulators has been demonstrated in a variety of materials and is an emerging pathway for next-generation spin-based computing. To modulate spin transport in these systems, one typically applies a sufficiently strong magnetic field to allow for deterministic control of magnetic order. Here, we make use of the well-known multiferroic magnetoelectric, BiFeO_{3}, to demonstrate nonvolatile, hysteretic, electric-field control of thermally excited magnon current in the absence of an applied magnetic field. These findings are an important step toward magnon-based devices, where electric-field-only control is highly desirable.