We develop a frequency-domain linear-response treatment of atomic photoionization within time-dependent density-functional theory, incorporating orbital-dependent self-interaction correction (SIC) through the optimized effective potential method in the Krieger-Li-Iafrate approximation. The first-order variation of the SIC potential is included in the self-consistent induced response. For Ar, SIC improves the subshell ionization thresholds and the 3p background cross section, mainly through the stationary reference and its associated Hartree and exchange-correlation response. The explicit SIC response has a smaller effect on the cross section but appreciably modifies the 3s-3p relative atomic delay near the 3s Amusia–Cooper minimum, with only moderate changes in the 3s cross section and little shift in the minimum position. Near 42 eV, the relative atomic delay for emission along the polarization axis changes from -243 as without SIC to -284 as with full SIC, approaching the experimental minimum of approximately -290 as. SIC also modifies relative Wigner delays near the outer ns photoionization minima of Kr and Xe. These results show that the stationary SIC reference and induced SIC response make distinct contributions to photoionization cross sections and attosecond delays.