We study theoretically the coherent nonlinear optical response of doped quantum wells with several subbands. When the Fermi energy approaches the exciton level of an upper subband, the absorption spectrum acquires a characteristic double-peak shape originating from the interference between the Fermi-edge singularity and the exciton resonance. We demonstrate that, for off-resonant pump excitation, the pump-probe spectrum undergoes a striking transformation, with a time-dependent exchange of oscillator strength between the Fermi-edge singularity and exciton peaks. This effect originates from the many-body electron-hole correlations which determine the dynamical response of the Fermi sea. © 2000 The American Physical Society.