The interaction of excitons with optical phonons is a significant contribution to the linewidth of excitons in quantum wells at room temperature. An electric field, applied in the direction of confinement, polarizes these exciton states, and their interaction with optical phonons is enhanced in polar semiconductors. In this paper, I investigate how this contribution to the linewidth of the electron--heavy-hole exciton changes with applied electric field for different sets of quantum-well parameters where the electron and heavy-hole subbands are taken to be parabolic.