Characterizing subglacial water flow is critical for understanding basal sliding and processes occurring under glaciers and ice sheets. Development of subglacial numerical models and acquisition of water pressure and tracer data have provided valuable insights into subglacial systems and their evolution. Despite these advances, numerical models, data conditioning, and uncertainty quantification are difficult, principally due to high number of unknown parameters and expensive forward computations. In this study, we aim to infer the properties of a subglacial drainage system in two dimensions using a framework that combines physical and geostatistical processes. The methodology is composed of three main components: (i) a channel generator to produce networks of the subglacial system, (ii) a physical model that computes water pressure and mass transport in steady state, and (iii) Bayesian inversion in which the outputs (pressure and tracer-transit times) are compared with synthetic data, thus allowing for parameter estimation and uncertainty quantification. We evaluate the ability of this framework to infer the subglacial characteristics of a synthetic ice sheet produced by a physically complex deterministic model, under different recharge scenarios. Results show that our methodology captures expected physical characteristics for each meltwater supply condition, while the precise locations of channels remain difficult to constrain. The framework enables uncertainty quantification, and the results highlight its potential to infer properties of real subglacial systems using observed water pressure and tracer-transit times.