The frequency response function (FRF) provides an input–output model that describes the system dynamics. Learning the FRF of a mechanical system can facilitate system identification, adaptive control, and condition-based health monitoring. Traditionally, FRFs can be measured by off-line experimental testing, such as impulse response measurements via impact hammer testing. In this paper, we investigate learning FRFs from operational data with a nonlinear regression approach. A regression model with a learned nonlinear basis is proposed for FRF learning for run-time systems under dynamic steady state. Compared with a classic FRF, the data-driven model accounts for both transient and steady-state responses. With a nonlinear function basis, the FRF model naturally handles nonlinear frequency response analysis. The proposed method is tested and validated for dynamic cutting force estimation of machining spindles under various operating conditions. As shown in the results, instead of being a constant linear ratio, the learned FRF can represent different mapping relationships under different spindle speeds and force levels, which accounts for the nonlinear behavior of the systems. It is shown that the proposed method can predict dynamic cutting forces with high accuracy using measured vibration signals. We also demonstrate that the learned data-driven FRF can be easily applied with the few-shot learning scheme to machine tool spindles with different frequency responses when limited training samples are available.