Magnetorheological polishing (MRF) has emerged as a critical non-contact sub-aperture polishing technology due to its notable attributes of high precision and minimal damage. However, MRF's inherent D-shaped removal function leads to reduced convergence efficiency of surface form error and introduces mid-spatial-frequency (MSF) waviness. To address these challenges, we propose magnetorheological precession finishing (MRPF) technology, which ingeniously combines MRF with bonnet precession polishing to generate a Gaussian-like removal function. A pivotal component of what we believe to be a novel approach is the design and fabrication of a specialized hemispherical magnetorheological precession polishing head. The design process incorporates magnetostatic simulations and magnetic force analysis to determine the optimal generating conditions for magnetorheological ribbons. Spot polishing experiments confirm the suitability of a 30° precession angle. Experimental results demonstrate that 8-step polishing achieves a Gaussian-like removal function. Additionally, uniform polishing of fused quartz surfaces significantly reduces Ra from 0.7 µm to 2.14 nm. This research showcases the feasibility of MRPF as a new technical route to achieve Gaussian-like removal functions and nanometer-scaled surface roughness.