Theophylline, a typical representative of active pharmaceutical ingredients, was selected to study the characteristics of experimental and theoretical solubility measured at 25 °C in a broad range of solvents, including neat, binary mixtures and ternary natural deep eutectics (NADES) prepared with choline chloride, polyols and water. There was a strong synergistic effect of organic solvents mixed with water, and among the experimentally studied binary systems, the one containing DMSO with water in unimolar proportions was found to be the most effective in theophylline dissolution. Likewise, for NADES, the addition of water (0.2 molar fraction) resulted in increased solubility compared to pure eutectics, with the highest solubilisation potential offered by the composition of choline chloride with glycerol. The ensemble of Statistica Automated Neural Networks (SANNs) developed using intermolecular interactions in pure systems has been found to be a very accurate model for solubility computations. This machine learning protocol was also applied as an extensive screening for potential solvents with higher solubility of theophylline. Such solvents were identified in all three subgroups, including neat solvents, binary mixtures and ternary NADES systems. Some methodological considerations of SANNs applications for future modelling were also provided. Although the developed protocol is focused exclusively on theophylline solubility, it also has general importance and can be used for the development of predictive models adequate for solvent screening of other compounds in a variety of systems. Formulation of such a model offers rational guidance for the selection of proper candidates as solubilisers in the designed solvents screening.