Natural Eutectic Solvents (NES) are a promising class of environmentally friendly liquids that offer an alternative to conventional organic solvents. To investigate their potential for extracting phenolic compounds from plant materials, we employed the COSMO-RS (COnductor like Screening MOdel for Real Solvents) computational method. Nineteen NES solvents were prepared to determine the most selective and effective solvent for extracting phenolic compounds from Teucrium chamaedrys using both theoretical and experimental evaluation. Screening of extraction efficiency was performed by quantification of individual compounds using ultra-high-performance liquid chromatography with a diode array detector and a triple-quadrupole mass spectrometer (UHPLC-DAD-MS/MS), as well as spectrophotometric assays (total phenolic content, total flavonoid content, and radical scavenging activity). In this paper, we propose a new approach to model NES properties using COSMO-RS. While the majority of previous studies have modeled these solvents as individual or pseudo-component complexes between hydrogen bond donor (HBD) and hydrogen bond acceptor (HBA), we have developed a ternary HBD:HBA:water supramolecular complex model to represent NES solution structures more accurately. This model provides insights into the intermolecular interactions driving the extraction process and predicts extraction efficiencies that agree with experimental data. Two eutectic mixtures (choline chloride:succinic acid in molar ratio 1:1 and 20% of water (w/w) and choline chloride: glycerol in molar ratio 1:1 and 20% of water (w/w)) showed strong affinity towards phenolic compounds. Overall, our findings suggest that the ternary complex model is a more appropriate approach for modeling NES properties.