Accurate estimation and forecasting of reference evapotranspiration (ETo) are essential for irrigation demand estimation. The FAO-56 Penman–Monteith formulation remains the physical standard for ETo computation, while recent advances in deep learning (DL) have demonstrated strong predictive skill for ETo forecasting. However, real-time ETo forecasting remains constrained by manual meteorological station identification, heterogeneous data acquisition, and labor-intensive preprocessing workflows. Existing software tools primarily support physics-based ETo estimation without real-time data integration or forecasting capability, whereas DL-based approaches often require manual data preparation, limiting automation and real-time applicability. This study introduces NeuralFAO56 Python package, a hybrid physics–data DL computational framework that embeds neural network–driven forecasting architectures for on-demand ETo estimation and forecasting across the continental United States (CONUS). NeuralFAO56 couples physics-based FAO-56 with DL sequence modeling within a unified pipeline that enables automated data acquisition, standardized preprocessing, and scalable deployment. The framework operates in dual modes: (i) physics-based FAO-56 ETo estimation using observed and forecasted meteorological inputs, and (ii) data-driven ETo forecasting using Long Short-Term Memory (LSTM) and Transformer architectures for multi-horizon (up to 7-day lead time) real-time forecast. The framework is evaluated across 867 stations at continental US spanning different climate regions. Results demonstrate strong short-term predictive skill, with performance degradation at longer lead times driven by reduced temporal predictability. Higher forecasting skill is observed in climatologically stable regions, while comparatively lower performance occurs in humid, convectively active regions. Overall, NeuralFAO56 provides a scalable, real-time framework that integrates physically based ETo modeling grounded in energy and mass conservation with DL forecasting and automated meteorological data pipelines to support short- to medium-range irrigation planning and management.