$\textbf{Purpose:}$ To develop and evaluate an operator learning framework for nonlinear inversion (NLI) of brain magnetic resonance elastography (MRE) data, which enables real-time inversion of elastograms with comparable spatial accuracy to NLI. $\textbf{Materials and Methods:}$ In this retrospective study, 3D MRE data from 61 individuals (mean age, 37.4 years; 34 female) were used for development of the framework. A predictive deep operator learning framework (oNLI) was trained using 10-fold cross-validation, with the complex curl of the measured displacement field as inputs and NLI-derived reference elastograms as outputs. A structural prior mechanism, analogous to Soft Prior Regularization in the MRE literature, was incorporated to improve spatial accuracy. Subject-level evaluation metrics included Pearson's correlation coefficient, absolute relative error, and structural similarity index measure between predicted and reference elastograms across brain regions of different sizes to understand accuracy. Statistical analyses included paired t-tests comparing the proposed oNLI variants to the convolutional neural network baselines. $\textbf{Results:}$ Whole brain absolute percent error was 8.4 $\pm$ 0.5 ($μ'$) and 10.0 $\pm$ 0.7 ($μ''$) for oNLI and 15.8 $\pm$ 0.8 ($μ'$) and 26.1 $\pm$ 1.1 ($μ''$) for CNNs. Additionally, oNLI outperformed convolutional architectures as per Pearson's correlation coefficient, $r$, in the whole brain and across all subregions for both the storage modulus and loss modulus (p < 0.05). $\textbf{Conclusion:}$ The oNLI framework enables real-time MRE inversion (30,000x speedup), outperforming CNN-based approaches and maintaining the fine-grained spatial accuracy achievable with NLI in the brain.