Multi-taper estimators provide low-variance power spectrum estimates that can be used in place of the windowed discrete Fourier transform (DFT) to extract speech features such as mel-frequency cepstral coefficients (MFCCs). Even if past work has reported promising automatic speaker verification (ASV) results with Gaussian mixture model-based classifiers, the performance of multi-taper MFCCs with deep ASV systems remains an open question. Instead of a static-taper design, we propose to optimize the multi-taper estimator jointly with a deep neural network trained for ASV tasks. With a maximum improvement on the SITW corpus of 25.8% in terms of equal error rate over the static-taper, our method helps preserve a balanced level of leakage and variance, providing more robustness.