Signaling bias refers to G protein-coupled receptor ligand ability to preferentially activate one type of signal over another. Bias to evoke signaling as opposed to sequestration has been proposed as a predictor of opioid ligand potential for generating tolerance. Here we measured whether delta opioid receptor agonists preferentially inhibited cyclase activity over internalization in HEK cells. Efficacy (τ) and affinity (KA) values were estimated from functional data and bias was calculated from efficiency coefficients (log τ/KA). This approach better represented the data as compared to alternative methods that estimate bias exclusively from τ values. Log (τ/KA) coefficients indicated that SNC-80 and UFP-512 promoted cyclase inhibition more efficiently than DOR internalization as compared to DPDPE (bias factor for SNC-80: 50 and for UFP-512: 132). Molecular determinants of internalization were different in HEK293 cells and neurons with βarrs contributing to internalization in both cell types, while PKC and GRK2 activities were only involved in neurons. Rank orders of ligand ability to engage different internalization mechanisms in neurons were compared to rank order of E max values for cyclase assays in HEK cells. Comparison revealed a significant reversal in rank order for cyclase E max values and βarr-dependent internalization in neurons, indicating that these responses were ligand-specific. Despite this evidence, and because kinases involved in internalization were not the same across cellular backgrounds, it is not possible to assert if the magnitude and nature of bias revealed by rank orders of maximal responses is the same as the one measured in HEK cells.