Debris-covered glaciers are increasingly studied because it is assumed that debris cover extent and thickness could increase in a warming climate, with more regular rockfalls from the surrounding slopes and more englacial meltout material. Debris energy-balance models have been developed to account for the melt rate enhancement/reduction due to a thin/thick debris layer, respectively. However, such models require a large amount of input data that are not often available, especially in remote mountain areas such as the Himalaya, and can be difficult to extrapolate. Due to their lower data requirements, empirical models have been used extensively in clean glacier melt modelling. For debris-covered glaciers, however, they generally simplify the debris effect by using a single melt-reduction factor which does not account for the influence of varying debris thickness on melt and prescribe a constant reduction for the entire melt across a glacier. In this paper, we present a new temperature-index model that accounts for debris thickness in the computation of melt rates at the debris-ice interface. The model empirical parameters are optimized at the point scale for varying debris thicknesses against melt rates simulated by a physically-based debris energy balance model. The latter is validated against ablation stake readings and surface temperature measurements. Each parameter is then related to a plausible set of debris thickness values to provide a general and transferable parameterization. We develop the model on Miage Glacier, Italy, and then test its transferability on Haut Glacier d’Arolla, Switzerland. The performance of the new debris temperature-index (DETI) model in simulating the glacier melt rate at the point scale is comparable to the one of the physically based approach, and the definition of model parameters as a function of debris thickness allows the simulation of the nonlinear relationship of melt rate to debris thickness, summarised by the Østrem curve. Its large number of parameters might be a limitation, but we show that the model is transferable in time and space to a second glacier with little loss of performance. We thus suggest that the new DETI model can be included in continuous mass balance models of debris-covered glaciers, because of its limited data requirements. As such, we expect its application to lead to an improvement in simulations of the debris-covered glacier response to climate in comparison with models that simply recalibrate empirical parameters to prescribe a constant across glacier reduction in melt.